Holik, Sonia Maria
Application of effective medium theory to the analysis of integrated circuit interconnects.
PhD thesis, University of Glasgow.
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The design and physical verification of contemporary integrated circuits is a challenging task due to their complexity. System-in-Package is an example of generally congested electronic components and interconnects which in the initial design process rely on computationally intensive electromagnetic simulations. Hence the available computer memory capacity and computational speed become meaningful limitations. An alternative method which allows the designer to overcome or reduce the limits is desired.
This work represents the first demonstration of the application of effective medium theory to the analysis of those segments of the entire integrated system where the interconnect networks are more dense. The presented approach takes advantage of the deep subwavelength characteristic of interconnect structures. In order to achieve the aim of defining the homogeneous equivalent for the interconnect grating structure a few steps were followed towards proving the homogenisation concept and finally presenting it by an analytical formulation. A set of parameters (metal fill factor, aspect ratio, dielectric background and period-to-wavelength ratio) with values related to typical design rules were considered. Relating these parameters allows the empirical models to be defined. In order to show the relationship between existing effective medium theories and those developed in this Thesis, the presented empirical models are defined in terms of the Maxwell-Garnett mixing rule with an additional scaling factor. The distribution of the scaling factor was analysed in terms of the calculated reflection and transmission coefficients of the homogenised structures that are equivalent to a given grating geometry. Finally the scaling factor, for each empirical model, was expressed by an analytical formula and the models validated by their application to the numerical analysis of grating structures.
The numerical validation was carried out by comparing the reflection and transmission coefficients obtained for the detailed and homogenised structures. In order to ensure the empirical models can be broadly employed, the performance of the model in the presence of non-normally incident plane wave was evaluated. For the range of angles ±30º the model is accurate to 5%. The impact of the shape of the grating, specifically the case of a tapered profile, typical of actual fabricated interconnects was also considered, with sidewall tapers of up to 5º giving the same error not higher than 5%.
Experimental validation of the application of the homogenisation concept to the analysis of interconnects is desired for two main applications: for the reflectivity estimation of a whole chip in a System-in-Package and for the performance estimation of interconnects on lower metal layers in an interconnect stack. For the first, free-space measurements are taken of a grating plate with copper rods aligned in parallel illuminated by a plane wave in the X-band (8.2-12.4 GHz). For the second, S-parameters are measured for microstrip waveguides with a number of metal rods embedded in the substrate between the signal line and ground plane. The good agreement with the simulations validates the homogenisation approach for the analysis of interconnects.
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