David Dahl

A Rigorous Approach for the Modeling of Through-Silicon-Via Pairs Using Multipole Expansions

This paper presents a novel multipole expansion method for the rigorous analysis of wave propagation along a through-silicon-via (TSV) pair. TSV models are often constructed using equivalent circuits under the assumption that the interfaces between different media are equipotential. In comparison with that, the proposed method is a rigorous full-wave solution of the propagation mode. It takes advantage of cylindrical wave expansion functions and matches the boundary conditions at both the metal-to-insulator and insulator-to-silicon interfaces exactly. The method is validated against a full-wave finite-element method solver, and analysis examples using the method are demonstrated in a frequency range up to 100 GHz.

Efficient Computation of Localized Fields for Through Silicon Via Modeling Up to 500 GHz

This paper presents methods for the modeling of the localized (near) fields of vertical interconnects in silicon interposers and the applications of these methods for the efficient computation of the electromagnetic properties of through silicon via structures. The localized fields are due to the mode conversions of the coaxial-to-radial waveguide junctions present in these structures. Because exact analytical techniques exist only for the homogeneously filled junction, an efficient numerical technique is proposed in this paper for the modeling of the inhomogeneous cases. This technique provides accurate results in the form of network parameters with three ports, which can be applied, e.g., in the framework of the physics-based via models. The finite-difference frequency domain method for the case of rotational symmetry is adapted to variable grid distances along the axial and radial coordinates, and interface conditions for the inhomogeneous filling of silicon and electrically isolating silicon dioxide are implemented. The method is validated with full-wave results from finite-element simulations and with the results from the published analytical methods that are adapted to the layered structures. The main focus is in the modeling for signal integrity analysis from the frequencies where the skin effect is well developed at about 100 MHz up to 100 GHz. Nevertheless, good agreement with the results from finite-element simulations up to 500 GHz is obtained for several relevant example structures, and a speedup of at least two orders compared with the finite-element simulations is achieved.

Applying a physics-based via model for the simulation of Through Silicon Vias

This paper presents a first approach for the efficient modeling of Through Silicon Vias based on a Physics-Based Via model for application in silicon interposers with metallic boundaries.

Efficient Total Crosstalk Analysis of Large Via Arrays in Silicon Interposers

In this paper, we present for the first time a rigorous crosstalk analysis of through silicon via (TSV) arrays consisting of several hundreds of TSVs in interposers with metallized surfaces, using the physics-based via (PBV) modeling approach for applications up to 500 GHz. The PBV modeling approach is valid for complete and almost complete metallizations of the substrate where radial wave propagation in the parallel-plate structure dominates the electromagnetic properties and is utilized with models of good accuracy for localized and propagating fields in the inhomogeneous dielectrics. The approach shows very good to good agreement of crosstalk results for frequencies up to 500 GHz in comparison to full-wave simulations and attains a speedup of at least two orders of magnitude in comparison to general-purpose simulators. The definition of a weighted power sum for total uncorrelated crosstalk is applied for all channels in the TSV array. These power sum results give more meaningful insights into the global effects of the parameter variations than single crosstalk contributions. Based on variations of several technology and design parameters of TSVs, we derive quantitative estimations of the impact of these parameters on the total crosstalk.

Effect of layered media on the parallel plate impedance of printed circuit boards

This paper evaluates the impact of layered media on the parallel plate impedance of printed circuit boards. In the parallel plate impedance computation using the radial waveguide method (RWM) for infinite planes and a contour integral method (CIM) for finite planes, the layered nature of investigated structures can be accounted for by effective wave numbers. Here, the effective wave numbers are obtained using either an exact solution from the transverse resonance method (TRM) or adopting approximations based on the assumption of quasi-TEM fields and the applicability of suitable averages. The wave numbers obtained with these techniques are compared for typical structures of interest, and the parallel plate impedance computed with effective wave numbers is evaluated and compared to a full wave solution. The results show that a comparatively simple approximation for the effective wave number is sufficient for an accurate parallel plate impedance calculation.

Modeling of differential striplines in segmented simulation of printed circuit board links

The decomposition, i.e., segmentation, of multilayer printed circuit board links into constitutive parts by means of network parameter representations is widely accepted. This work examines implications of this procedure regarding differential links. For the first time, the physics-based single-ended stripline port is generalized to model coupled striplines in the environment of multilayer PCBs. Notably, results are general in nature and are not limited to the employed methods. The proposed approach allows for segmentation and computationally efficient simulation of links between very large via pin fields, where differential stripline pairs pose a common routing scheme. Its feasibility is validated using self-contained physics-based simulations as well as measurements and full-wave simulation results up to 40 GHz.

High Frequency Characterization of Silicon Substrate and through Silicon Vias

In this work, we present the high frequency extraction of electrical material properties of silicon substrates. Two methods, including the substrate integrated waveguide (SIW) based method and the planar resonator based method, are used and their consistency will be shown. For the SIW-based method, a line difference algorithm is applied for the calculation of a broadband material property with the effective width of the SIW determined experimentally. For the planar resonator based method, the material parameters are extracted at discrete resonance frequencies by comparing to full-wave simulations. The results from both methods are compared to each other and discussed with respect to their accuracy and their suitability for different substrate types. For the validation purpose, TSV transitions are designed and fabricated on the same wafers of SIWs and planar resonators, which are modelled using a full-wave electromagnetic solver together with the extracted electrical properties of the silicon wafer. The comparison of the modelling to the measurement results will be presented and the influence of the variation of substrate material properties on the high frequency TSV performance and the accuracy of TSV modelling are discussed.