Application of the transverse resonance method for efficient extraction of the dispersion relation of arbitrary layers in silicon interposers

Abstract

In this article, the dispersion relation for horizontal wave propagation in silicon interposers consisting of arbitrary numbers of silicon and silicon dioxide layers between metal layers is investigated. Results are obtained by the transverse resonance method (TRM). The method is verified by comparison to results of finite element based full-wave simulations. The results of the TRM show good correspondence with those obtained by full-wave simulations and can be obtained within significantly shorter calculation times. Results for a typical layout of a silicon interposer and examples for thin silicon dioxide layers and a large number of layers are also presented. Further, an approximate solution for the fundamental mode of typical structures is given and examples show the applicability and limitations of the approximate solution

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

“…1, the fundamental mode in the layered waveguide structure is transverse magnetic (both with respect to the direction of propagation and the direction of layering). A suitable method for the computation of the exact wave number of this mode is the transverse resonance method (TRM) [11], [12], which numerically solves an analytical expression for the fields corresponding to the fundamental mode of a structure. Alternatively, approximations based on a direct analytic approach can be employed.…”

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“…Of the two fundamental modes which are supported by this layered structure only the fundamental TM mode needs to be considered. More details and further references can be found in a preceding paper [7]. By solving the transcendental equation which arises from the matching of the tangential field components at the interfaces, the exact wave number of the structure is obtained.…”

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“…Further physical justification is offered by a seminal paper by Marcuvitz and Schwinger [2] where we find a first version of the transverse resonance condition as a means to satisfy certain boundary conditions via appropriate impedance matching. Extensive use of the telegrapher's equation is also found in Ramachandran [3], Gallawa [4], Clarricoats and Oliner [5,6], Yoneyama [7], Borneman and Arndt [8], Tao [9], Shigesawa and Tsuji [10] as well as Dahl et al [11]. More recently, Moshonas et al [12] also used transmission line methods for the study of photovoltaic cells.…”