Summary and Conclusions Regarding Measurements

After a comprehensive discussion of the design of the test structures, the simulation setups, and the most relevant aspects of the configuration of the simulations, several results for measurements have been presented. It could be observed that despite several uncertainties regarding exact parameter values, overall, a good to fair agreement could be obtained in the reference simulations. Using silicon substrates of relatively high resistivity (here about 1 S/m), links can be established which should be transparent enough for the application in an interposer.

For future test structure designs and measurements, a vertical probing similar to the one presented in [128] could be evaluated. Implementation of stackups similar to the ones used in the model approach comparison of Section 7.2 could be implemented as physical test structures to further justify the assumptions made and validate the modeling approaches.

The application of a substrate bias voltage could be used to further investigate the depletion layer influence.

7.3 Correlation of Measurements and Full-Wave Simulations

(a) (b)

(c) (d)

Figure 7.16:Measurement and simulation results for structure V10 (→Fig. 7.10a, Table H.1) on wafer W23B (→ Fig. 7.2). Measurement 1 uses setup 1b, Measurement 2 uses setup 1a (→Table H.4). All simulations use a silicon conductivity of1 S/m, a silicon layer thickness of 80µm, and a thickness of the WPR layer of3.5µm. The thickness of the oxide layer is200 nm in Simulation 1, 150 nmin Simulation 2, and 250 nmin Simulation 3.

(a) (b)

(c) (d)

Figure 7.17:Measurement and simulation results for structure V13 (→Fig. 7.10a, Table H.1) on wafer W23B (→ Fig. 7.2). Measurement 1 uses setup 2b, Measurement 2 uses setup 2a, and Measurement 3 uses setup 1b (→ Table H.4). All simulations use a silicon conductivity of 1 S/m and a silicon dioxide layer thickness of 500 nm. Simulation 1 uses a thickness of the WPR layer of 3.5µm and a thickness of the silicon layer of 80µm. Simulation 2 uses a thickness of the WPR layer of4.5µmand a thickness of the silicon layer of100µm.

7.3 Correlation of Measurements and Full-Wave Simulations

(a) (b)

(c) (d)

Figure 7.18:Measurement and simulation results for structure V5 (→Fig. 7.10a, Table H.1) on wafer W23B (→Fig. 7.2). Measurement 1 uses setup 2b, Measurements 2 and 3 use setup 1b (→ Table H.4). All simulations use a silicon conductivity of 1 S/mand a silicon dioxide layer thickness of 500 nm. Simulation 1 uses a thickness of the WPR layer of 3.5µm and a thickness of the silicon layer of 80µm. Simulation 2 uses a thickness of the WPR layer of 4.5µmand a thickness of the silicon layer of100µm.

(a) (b)

(c) (d)

Figure 7.19:Measurement and simulation results for structure V13 (→Fig. 7.10a, Table H.1) on wafer W09 (→Fig. 7.2). Measurement 1 uses setup 1a(→Table H.4). All simulations use a thickness of the WPR layer of4.5µm, a thickness of the silicon layer of100µm, and a silicon dioxide layer thickness of500 nm. Simulation 1 uses a silicon conductivity of2×10³S/mand Simulation 2 uses a silicon conductivity of1×10⁵S/m.

7.3 Correlation of Measurements and Full-Wave Simulations

(a) (b)

(c) (d)

Figure 7.20:Measurement and simulation results for structure V13 (→Fig. 7.10a, Table H.1) on wafer W10A (→ Fig. 7.2). Measurement 1 uses setup 2b, Measurements 2 and 3 use setup 1b (→Table H.4). Simulation 1 uses a silicon conductivity of2×10³S/m, a thickness of the WPR layer of 4.5µm, a thickness of the silicon layer of 100µm , and a silicon dioxide layer thickness of180 nm.

Chapter 8

Conclusion and Outlook

This thesis has presented contributions to the electromagnetic modeling of through silicon vias (TSVs) for application in silicon interposers. It has been shown that by adaptations of both the near field and the far field parts of the model, a physics-based via model that was originally conceived for vias in multilayer printed circuit boards achieves a very good accuracy up to several hundreds of Gigahertz in a large parameter range. In addition, the already high numerical efficiency of the method could be increased further.

The method has proven itself to be applicable even with metals of lower conductivity such as aluminum and with incomplete metallizations. Major limitations have only been found in cases with very narrow spacing of the TSVs. The relevant dimensions have been determined to be the minimal distances between antipad clearances and the cavity height. For the near field modeling, a finite-difference frequency-domain method has been developed and shown to lead to results of high accuracy. The rotational symmetry that is exploited during the computation of the near field leads to a high numerical efficiency. It is therefore the recommended method for near field computation at all frequencies.

The overall reduced computational effort of the modeling has enabled the analysis of the crosstalk of large via arrays. The power sum of total uncorrelated crosstalk has been justified as a parameter for the efficient estimation of effective crosstalk levels in first design phases of large TSV arrays. The summation in terms of power has been extended to periodic digital signals and enables the reduction to a single figure of merit for each via to quantify a typical crosstalk value. Overall, the presented methods constitute a numerically very efficient procedure for the analysis and design of large and therefore realistic arrays.

Complementary modeling approaches have been motivated, discussed and correlated with the PBV models for the modeling of TSVs in several stackup environments. Even though an influence of the stackup environments could be observed, it has been found that the general frequency behavior and the orders of magnitude of transmission and crosstalk are similar.

In the last part of this thesis, measurement and FEM full-wave simulation results have been

correlated for several structures that aim to characterize electromagnetic characterization of TSVs. A good agreement could be observed which validates the applicability of both the FEM-simulations and the FEM-models.

In future research on the modeling of TSVs, the PBV model could be extended in order to achieve an even large parameter range where it is applicable. Especially a model for the near field coupling between TSVs is of interest. At several hundreds of Gigahertz, it might also be required to take into account other effects that have been neglected so far such as the surface roughness of conductors. Regarding the simulation of interposers with partial metallizations, criteria could be developed to decide which of the modeling approaches, the physics-based model or the multi-conductor model, is better suited for an interposer inside a specific electromagnetic environment. The measurements have shown that a good transmission over TSV transitions is possible if silicon substrates of high resistivity are used. Therefore, the application of an interposer in a demonstrator for signal transmission at several tens of gigabit could be feasible.

Appendix A

Numerical Methods

A.1 Root Search Techniques for the Complex Domain

The analytical methods for the determination of the wave number of modes in several guiding structures presented in this thesis can be formulated as the mathematical problem of finding locations with function values of zero in the complex plane of a single valued complex function. The chosen methods to solve this task are typically a tradeoff between several factors such as the generality of application, the overall numerical efficiency, and the ease of implementation.

First in this chapter, the Newton-Method and related techniques are discussed. These method are based on a good initial guess of the root value and under certain conditions enable an iterative approximation towards the true value. As the functions encountered with the presented waveguide problems have an infinite number of complex roots and guessing the initial values that lead to a converging series of iterations towards the desired root is difficult, additional techniques are in many cases helpful. The argument principle method for the generation of these start values from the evaluation of contours of areas in the complex domain is therefore also presented in this chapter.