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An empirical effective medium approximation that provides a homogeneous equivalent for a layer of interconnects un-derneath a spiral inductor is presented. When used as part of a numerical 3D model of the inductor, this approach yields a faster simulation that uses less memory, yet still predicts the quality factor and inductance to within 1%. We expect this technique to find use in the electromagnetic modeling of System-on-Chip.

On-chip spiral inductors are usually fabricated over areas that are either unused or contain dummy fill and/or patterned ground shields. While fabrication process geometries continue to shrink, inductor sizes are fixed by the usual physical laws. Hence, the use of passive components in System-on-Chip is increasingly inefficient. The under-utilised area underneath inductors could alternatively be used for routing interconnects associated with other circuits (e.g. buses) resulting in significant downscaling of the size of the entire chip. Unfortunately, this presents a challenging electromagnetic modelling problem that is difficult to solve with existing numerical techniques. In particular, the number of features necessary to adequately represent interconnects below an inductor is large due to the small geometrical size of the wires compared to the area occupied by the inductor. Such simulations can be done on isolated structures [

A numerical model of an example inductor was built and solved using HFSS software [^{6} S/m. The underpass at Metal 2 has thickness 0.74 μm. The isolating silicon-oxide layer between Metal 2 and Metal 3 is 0.8 μm thick, while between Metal 2 and Metal 1, and Metal 1 and the substrate, it is 1 μm thick. The interconnects are placed on Metal 1, and are defined as floated metal (copper) rods, i.e. they are not connected to ground. The rods are arranged in parallel, and have length 300 μm in order to cover the entire span of the inductor. The rods have width 0.45 μm and height 0.9 μm giving an aspect ratio of 2. The space between the rods is 0.45 μm, giving a metal fill factor of 0.5. The rods and inductor were modelled using layered impedance boundary conditions. The substrate is defined as 15 Ω-cm resistivity silicon of thickness 625 μm and is grounded by the perfect electric boundary conditions defined at the bottom plane, while other walls were defined as radiation boundaries. The spiral was covered by a 60 nm thick silicon-nitrate passivation layer.

A modified Maxwell-Garnett mixing rule [3,4] is used to calculate the effective permittivity of the homogeneous equivalent layer

where ε_{i} and ε_{e} are the dielectric functions of the inclusion and host material respectively (here, a metal and a dielectric), Y is a constant relating the fields inside and outside the inclusions (typically Y = 3 for spherical inclusions), f is the filling factor or ratio of the volume of the inclusion to the total size of the unit cell. The Maxwell-Garnett rule is known to give a qualitatively correct prediction of the effective properties of a composite with conducting inclusions [_{i} is expressed by a Drude model.

First, it was verified that placing metal rods underneath the spiral inductor influenced its electrical characteristics.

the value and frequency of the maximum Qfactor did not change, but the SRF dropped by 4% to 4.9 GHz (9% to 4.7 GHz). The more metal rods underneath the spiral, the higher the parasitic capacitance in turn causes lowering of the SRF. The insert plots the inductance and similarly indicates the SRF reduces when rods are present.

Next the rods were replaced with a homogenous equivalent layer of the same thickness. In the empirical fitting procedure the linearly distributed values of the scaling factor Ψ were sampled within the interval of 1 to 10000 with the step increment of 100. For the particular geometry arrangement denser sampling of the values of Ψ does not reflect in notable change in the characteristic inductor’s parameters. It was found that for the studied spiral inductor the fitted value of complex effective permittivity gives good approximation of both the Q-factor and inductance in the considered frequency range of 0.1 - 8 GHz when the coefficient Y = 7000 and gives ε_{eff} = 28004 + 7.4069 × 10^{–}^{3}i. The results from the EMT simulation of 50 rods are compared against the detailed model in _{eff} than either of the components, in certain mixing ratios. Applying the EMT technique gives significant reduction in computational time (47%) and allocated memory (45%) for the case of having 20 metal rods below inductor. This decrease in time and memory consumption approaches 48% when 50 rods are included and is predicted to decrease further as the complexity of the model increases.

A numerical study was used to demonstrate that a single layer of interconnects aligned underneath a spiral inductor can be replaced by a homogenous equivalent with effective permittivity calculated based on effective medium theory. The homogenised equivalent structure predicts the values of quality factor and inductance to within 1%. It is also demonstrated that the inductor’s performance is affected by the number of conducting rods placed below the spiral. Now that the concept has been demonstrated, future work involves calculating and tabulating

effective dielectric constant for a range of commonly uncounted geometries.