In the following, several TSV arrays in metal-clad interposer of medium sizes are simulated.
The simulations aim at the validation of the proposed PBV approaches by comparison with FEM full-wave simulation results [112]. The magnitudes of scattering parameters presented in the following show the crosstalk from one coaxial via port located on the top of the struc-ture to ports of other vias on the bottom side, i.e. the far-end crosstalk (FEXT) is shown exclusively. The scattering parameter are normalized to50Ω. The following investigations are limited to the FEXT because reflection and transmission have already been validated in Section 5.5 and far- and near-end crosstalk differ mainly in phase. The layout of the
6.2 Validation of the PBV Model for Mid-Scale TSV Arrays
Figure 6.1: Validation structures with port numbering for the signal via ports. The first numbers above each via refers to for the (coaxial) top ports, the second number to the (coaxial) bottom port. Ground vias are shown as simple circles. The default pitch between all neighboring vias is dpitch = 200µm. (a) case with infinite planes (b) case with finite planes with, if not stated otherwise,∆ρx = ∆ρy = dpitch. Figure and text taken from [14].
considered structures including the assignment of signal and ground vias is illustrated in Fig. 6.1. Table 6.1 gives the the default parameter values. Further, the respective parameter variations are given in the figure captions.
The termination of the appropriate termination of the (azimuthally) anisotropic modes has been summarized in Sec. 3.9. It has been stated that the short-circuit should be the termination of choice for most practical cases. To gain more insight into the relevance of the type of mode termination, results for mode terminations with open circuits are also shown in several of the following cases.
The validation results are presented in Figures 6.2 through 6.4. These cover variations of the silicon conductivity, of the via pitch, and of the substrate metallization plane sizes.
In general a good agreement of results can be observed in Figs. 6.2a and 6.2b. In some cases larger deviations are found at lower frequencies. There, a simple (inductive) behavior is expected [113]. Therefore, the variations of the reference results in these regions are unphysical.
As can be seen in the inset of Fig. 6.2a, below 600 MHz the reference simulation results show a more complex behavior than the PBV results. The latter are more reasonable since the crosstalk should show a simple inductive behavior at low frequency, cf. Appendix D.3.
Frequency (GHz)
Figure 6.2:Far-end crosstalk from port 1 to ports 4, 6, 8, 22, and 32 (order of decreasing level at 50 GHz) for the layout in Fig. 6.1a with default parameter values and a pitch of 200µm. Both PBV variations with either short-circuited or open-circuited ports of the anisotropic modes are shown. (a) With a silicon conductivity of10 S/m. (b) With a silicon conductivity of100 S/m. Figure and text adapted from [14].
6.2 Validation of the PBV Model for Mid-Scale TSV Arrays
Figure 6.3: Far-end crosstalk for the layout in Fig. 6.1a with the following port ordering corresponding to decreasing levels at 50 GHz. (a) From port 1 to ports 3, 4, 6, 8, 22, and 32 with reduced pitch of 100µm and silicon conductivity of 1 S/m. Inset: Detail of FEM reference results forS1,3 and S1,4 and the PBV results for bothS1,3 and S1,4. (b) From port 1 to ports 4, 8, and 32 with default pitch and a silicon conductivity of 10 S/m. A maximum deviation of4.8 dBand a mean deviation of0.42 dBare obtained forS1,4 with the PBV using
Description Name Default Value
Radius of the via barrel rvia 15µm
Radius of the clearance/antipad rantipad 30µm
Pitch dpitch 200µm
Minimal distance between via antipads dmin dpitch−2rantipad
Thickness of the oxide toxide 1µm
Thickness of the silicon tsilicon 100µm
Height of the cavity tcavity tsilicon+ 2toxide Thickness of the plane metallizations tmetal 5µm
Relative permittivity of silicon εr,Si 11.9 Relative permittivity of oxide (SiO2) εr,oxide 4
Table 6.1: Default Parameter Values for the Validation Setups. Table adapted from [14].
Above 5 GHz a maximum deviation of 0.28 dB and a mean deviation of 0.084 dB are ob-tained forS1,4 with the PBV using short-circuit terminations.
In the inset of Fig. 6.2b it can be observed that below4 GHzthe reference simulation results show again unphysical behavior. Above5 GHz a maximum deviation of1.5 dBand a mean deviation of 0.24 dB are obtained for S1,4 with the PBV using short-circuit terminations.
Good agreement with reference results is also obtained for most of the results from about 5 GHz to 100 GHz. The only major exceptions are those cases where a close spacing of vias in comparison to the cavity height is used. As discussed before, the PBV results are inaccurate in these cases because a near field coupling in addition to the coupling through parallel plate modes occurs. The existence of this mechanism can be concluded from the difference in near- and far-end crosstalk (S1,3 and S1,4 , respectively) as observed in Fig. 6.3a. The relevant parameters that should be compared are the cavity heighttcavity and the minimal distance between viasdmin. In case of signal vias the latter is the minimal distance between the antipad regions. The order of the errors of the PBV that can be expected for different values of the ratio minimal distance to cavity height dmin/tcavity should be discussed here: For a ratio dmin/tcavity = 1, a PBV error of about 0.5 dBand for a ratiodmin/tcavity = 0.5a PBV error of about3 dBare obtained in Fig. 6.3a. For a minimal distance between antipads of 40µm in Fig. 6.3a, a ratio of dmin/tcavity = 0.4 results in an error of about 4.6 dB starting at a few GHz.
The investigation of higher frequencies in Fig. 6.3b shows very good agreement up to 200 GHz and good agreement up to 500 GHz. Also for the results with finite planes in Fig. 6.4, where both the PBV and the FEM simulation use open circuited boundaries at the plane edges, good agreement is obtained.
In conclusion, there is a good general agreement of the proposed PBV method with the
6.2 Validation of the PBV Model for Mid-Scale TSV Arrays
Frequency (GHz)
0 50 100 150 200 250 300 350 400 450 500 Magnitudeofscatteringparameters(dB) FEM [112]
PBV, short
0 5 10
-100 -80 -60 -40
-100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0
Figure 6.4: Far-end crosstalk from port 1 to ports 4, 8, 6, 32, and 22 (order of decreasing level at50 GHz) for the layout in Fig. 6.1b with silicon conductivity of 10 S/m. Inset: For the case with finite planes a very good agreement with reference results can also be obtained for low frequencies. Figure taken and text adapted from [14].
FEM reference results in the results presented in Fig. 6.2a–6.4. In the cases of close spacing there are larger deviations with the FEXT being consistently overestimated.
6.2.1 Numerical Performance
The PBV simulations have been executed using implementations in MATLAB [114]. The computer system has an Intel Core i7-960 8-core 3.2 GHz CPU and 24 GB of RAM. For a TSV array that is significantly larger than the examples shown before, the magnitudes of scattering parameters are presented in Fig. 6.5. A3×3pattern is used with ground vias at two opposite corners and signal vias elsewhere. From this pattern, several arrays of different sizes are constructed and the admittance parameters are simulated with the isotropic and four anisotropic modes (modes with indices 0,±1,±2, cf. Appendix D). The simulation times per frequency point are given in Table 6.2. On the used computer system, only the smaller arrays in Table 6.2 can be simulated due to limitations in RAM. For the FEM results in Table 6.2, both the simulation time for the “adaptive process” at the solution
frequency (iterative mesh refinement until a convergence measure is below a certain limit) and the per-frequency-point simulation time during the consecutive sweep are presented.