In this section, near field models are used in the PBV model in conjunction with the far field models presented in the previous chapter to obtain network parameter descriptions of silicon interposers. A description is thereby obtained for via links assigned to single vias and referenced to their coaxial ports. These ports are illustrated as A2 and B2 in Fig. 5.1.
Three different variants are considered for the PBV modeling and listed in Table 5.1.
For the first variant, it is assumed that the near field model can be represented as in Fig. 3.6c, i.e., with only admittances which are parallel to each (coaxial) via port, and thus analogous to the Yeff in Fig. 3.6c. Qualitatively, it can be expected that this is valid for low frequencies. From the FDFD-admittance matrix Y, such an admittance Yap can be obtained. By identification of the entriesY1,1 =Yap+Y1,1(R) and Y1,2 =−Y1,1(R) one concludes
5.5 Application in Physics-Based Via Modeling Variant Description/Comment
1 Only parallel admittancesYap (cf. (5.27)) to model the near fields.
2 Segmentation of near fields as three-port at inner (barrel) radii.
3 As Variant 2, but with first anisotropic modes [56,94].
Table 5.1: Physics-Based Via Model variants in Figs. 5.11 to 5.15. Table adapted from [13].
that
Yap =Y1,1+Y1,2. (5.27)
The propagating field model uses only the isotropic mode for variants 1 and 2 and addi-tionally the first anisotropic modes for variant 3. As it is discussed in Section 3.5.1 and validated in Section 6.2, anisotropic modes are short-circuited for the propagating field model. A radial line description as presented in (4.34) is used for de-embeeding/extraction of a near field model from the FDFD results. This near field model is used in variants 2 and 3 for concatenation at the radial ports (ports C in Fig. 5.1). It has to be noted that a de-embedding is not strictly necessary as the radial ports of the CIM could also be adapted. Even if an FDFD domain size is chosen such that the corresponding CIM radial ports overlap, this would still be supported by the CIM method.
The geometry and results of the first application example are depicted in Fig. 5.11 in terms of magnitudes of scattering parameter results normalized to 50Ω. The structure has two signal and four ground vias. Due to the symmetry of the structure, only four of the overall 16 network parameters are of interest. As can be seen, the agreement of the proposed PBV methods show a good agreement with the FEM reference simulations [66].
FEM simulation have been carried out both with metal component of perfect conductivity and with metal component with the conductivity of aluminum of σ = 3.4×107S/m. No significant differences can be observed between these two result sets. Larger deviations between the different PBV model variants can be observed only for variant 1 compared to the other variants. The inclusions of anisotropic modes has a small impact as is apparent from the good agreement of variants 1 and 2.
As has been discussed before, the consideration of complete plane metallizations is one of the limiting cases that enables an efficient physics-base modeling. In order to give in insight into the applicability range two cases with plane perforations have been simulated using FEM full-wave simulations and the results have been compared to the PBV results without plane perforations. The two cases of plane perforations are named “Mod.1” and “Mod.2”
in Fig. 5.12 and can be described as follow: The first structure is named “Mod.1” and has comparatively large circular cutouts of only the top metallizations. The remainder of the structure (configuration of signal and ground vias) is the same as the as in Fig. 5.11. The
1
Figure 5.11:First application example with a pair of signal vias accompanied by four ground vias in an infinite parallel plate structure. In the reference FEM simulations both PEC and aluminum are considered as material for the via barrels and metallic plates. Dimensions are given in the insets, the silicon conductivity is10 S/m. The silicon permittivity is11.9·ε0 and the silicon dioxide permittivity is4·ε0. A typical simulation time per frequency point for the FEM is 30 s; a typical time for the proposed method is 0.3 s. Figure taken and text adapted from [13].
5.5 Application in Physics-Based Via Modeling
Figure 5.12: Second application example with a pair of signal vias accompanied by four ground vias as in Fig. 5.11 circular perforations. Via dimensions are the same as in Fig. 5.11, the silicon conductivity is0 S/m, silicon permittivity is11.9·ε0and silicon dioxide permittivity is 4·ε0. The geometry of the plane perforations is given in the insets. Variant “Mod.1” has circular cutouts in the top plane, variant “Mod.2” has circular cutouts in both planes. Typical simulation times per frequency point for the FEM are 15 minutes (Mod.1) and 10 minutes (Mod.2); a typical time for the proposed method is0.3 s. Figure and text adapted from [13].
Frequency (GHz)
Figure 5.13: Third application example with a pair of signal vias accompanied by four ground vias. The difference of the configuration compared to Fig. 5.11 is the closer spacing of the vias. The silicon conductivity is10 S/m, the silicon permittivity is11.9·ε0, and silicon dioxide permittivity is 4·ε0. Figure and text taken from [13].
5.5 Application in Physics-Based Via Modeling
Figure 5.14: Fourth application example with an additional, central ground via compared to Fig. 5.11. Dimensions are given in the insets, the silicon conductivity is10 S/m. Figure and text taken from [13].
Frequency (GHz)
Figure 5.15: Fifth application example: two signal vias with ports at both ends and two rows of ground vias (via fences). Dimensions are given in the insets, the silicon conductivity is10 S/m. Figure and text taken from [13].
5.5 Application in Physics-Based Via Modeling second structure is named “Mod.2” and has comparatively small cutouts. These cutouts are around and between the vias (which again have the same configuration as in Fig. 5.11) and have sizes comparable to the via dimensions. In this case, the cutouts are applied to both the top and the bottom metallizations. Because models for the cutouts are not available for the PBV, they are not considered in it. The agreement of the results is still good up to 150 GHz for variant “Mod.1” and to about 400 GHz for variant “Mod.2”. In the FEM simulations empty air boxes with absorbing boundary conditions are attached to the areas of the perforated planes to model the environment. The influence in practical environments consisting of, e.g., redistribution layers of an interposer is thereby ignored.
A prerequisite for the applicability of the PBV model as presented in the previous chapter is the coupling of vias only through a propagating field. This condition is met if the via pitch is large enough in comparison to the distance at which a coupling through cut-off higher order parallel plate modes occurs. This distance in lateral directions is dependent on several parameters. It is, e.g., proportional to the height of the cavity between the plane metallizations. For the PCB application of the PBV, it has already been observed in [65] that an additional near-field coupling should be included in order to accurately model cases of very close via spacing. Using FEM full-wave results, it is investigated in the example shown in Fig. 5.13 which influence reduced distances between vias have. In comparison to the structure in Fig. 5.11 only the distances between signal vias and to ground vias are reduced, all other parameters are the same. It can be observed that there is still a good agreement with the reference full-wave results for the transmission S1,1 and the reflectionS1,2. In contrast, the crosstalk parametersS1,3andS1,4 show larger deviations.
These deviations are most pronounced towards lower frequencies.
The next example is shown in Fig. 5.14 and allows to investigate the effect of an interme-diate ground via between two signal vias. Variant 1 of the PBV model shows again the strongest deviations for frequencies above 150 GHz. Variant 2 also shows stronger devi-ations from the reference results Variant 3. Therefore, in this case, a clear advantage of including the first anisotropic modes in the far field model can be observed.
In the last example, which is shown in Fig. 5.15, a structure that has some similarity with a substrate integrated waveguide is considered. The structure consists of two signal vias and several ground vias constituting two ground via fences. Using the PVB in Variants 2 and 3, a good agreement can be observed up to about 400 GHz
In summary, it can be conluded that this section proves that with the proposed adaptation of the PBV, i.e., by using a local FDFD-based full-wave modeling and the CIM with wave number for the layered structure, an accurate modeling technique is available. The accuracy has been validated using several relevant example structure. The best accuracy
can be obtained by using the complete near field model which is obtained through a de-embedding procedure and by using also the first anirostropic mode in the CIM for the far field model.