**1. Introduction**

For over four decades the evolution of electronic technology has followed Moore's low where the number of transistors in an integrated circuit (IC) approximately doubles every two years. This naturally increases the number of internal interconnections needed to complete the system. The increase in chip complexity is achieved by a combination of dimensional scaling and technology advances. A variety of chip types exist, including memory, microprocessors and application specific circuits such as System-on-Chip (SoC). Since it is expensive to fabricate the large and simple passive components such as on-chip capacitors and inductors on the same die as the active circuits, it is desirable to fabricate these on separate dies then combine them in System-in-Package (SiP). The main advantage of SiP technology is the ability to combine ICs with other components, including passive lumped elements already mentioned but also antennas, high speed chips for radio frequency communication etc., into one fully functional package. The high complexity of SiP brings many challenges to the design process and physical verification of the system. In many cases the design process relies on detailed 3-D numerical electromagnetic simulations that tend to be slow and computationally demanding [1,2,3] in many cases limited by the available computer memory capacity and computational speed. Therefore, directly including the detail of the dense interconnect networks into the numerical model is demanding due to the amount of memory required to hold the detailed mesh, and numerical penalties associated with small mesh cell sizes relative to the wavelengths of the signals being modelled.

From a package-level point of view, the on-chip interconnects can be seen as a mixture of metal inclusions located in a host dielectric [4-8]. Most of the current studies based on numerical analysis of 2-D or 3-D structures with two constituents show that the effective properties of the mixture strongly depend on the volume fraction, its geometrical profile

© 2012 Holik, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and spatial orientation in periodic or random arrangements [9,10]. It has been shown that the macroscopic properties of dielectric-only mixtures can be represented by a homogenous dielectric with an effective permittivity that is determined using an empirical mixing model [11]. Metal-dielectric mixtures have been less thoroughly explored, with work limited to treating spherical or ellipsoidal metal inclusions [12,13]. The approach was extended [4-8] to cope with rectangular cuboid metal inclusions representative of on-chip interconnect structures. The use of a single fitting parameter was retained, and is calculated for a wide range of aspect ratios (0.6 – 3), dielectric host materials (1 – 11.7), metal fill factors (0.2 – 0.6) and signal frequencies (1 – 10 GHz) that are likely to be of interest to System-in-Package designers. Here a simplified empirical mixing model defined for a narrowed down range of aspect ratios (1.4 – 3) which accounts for the interconnect geometries is presented. The model is at the same time straightforward and more accessible as well as more accurate. The accuracy improvement is related to the neglected range of low aspect ratios where the scaling factor has more rapid increment [4, 5].

Empirical Mixing Model for the Electromagnetic Compatibility Analysis of On-Chip Interconnects 201

2 *i e*

  (1)

(2)

*i e ie*

 

2

*p*

() 1 ( )

where is the frequency of interest, p is the plasma frequency and is a damping term representing energy dissipation. Despite wide spread use of copper interconnects for the intermediate levels of the interconnect stack, aluminium is often used for the global wiring with which we are concerned, and has p = 15 eV and = 0.1 eV [19]. Note that the energy is related to the free space wavelength 0 by = 1.24x10-6(0)-1. This model was used in both

Rigorous coupled wave analysis (RCWA) [20] was performed for the TM polarisation with the electric field vector **E**TM coplanar with the grating vector **K** as shown in Fig. 1. The structure was illuminated by a plane wave with incident angle -89 ≤ θ ≤ 89º and free-space wavelengths λ = 30 cm, λ = 10 cm, λ = 6 cm, and λ = 3 cm. Hence, the adjusted height h of the homogenised layer does not simplify the model the height of the homogenised equivalent layer was kept the same as the grating. The reflection and transmission coefficients for the homogenised structure were calculated using an analytical formula defined for a stratified

It is not necessary to make dependent on the host dielectric as this is already accounted for explicitly in Eq. (1). It was verified by numerical experiment. The scaling factor in the narrowed down range of aspect ratios was observed to have a linear dependence on this

 

*<sup>i</sup> <sup>j</sup>*

 

interconnect pitch is fixed = 100 μm for the sake of clarity in illustrating the method. While we include the effect of the interlevel dielectric, we deliberately neglect the bulk substrate

> *<sup>f</sup> <sup>f</sup>*

where i and e are the dielectric functions of the inclusion and host material respectively (here, a metal and a dielectric), is a constant relating the fields inside and outside the inclusions (typically = 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 [18]. In earlier work a limited example of such an approach for a fixed value of applicable to a single structure was presented [6-8]. Further, the approach was expanded by developing a compact equation to calculate the appropriate value of for a broad range of parameters [4,5]. Here a new empirical model is presented where the considered values of aspect ratio are within 1.4 – 3. The frequency dependent dielectric function of a metal inclusion i() can be expressed by a

*eff e e*

 

and all but the top layer of interconnects.

the analytical and numerical calculations.

medium comprising a stack of thin homogenous films [21].

parameter, therefore the general form of the empirical model was chosen:

Drude model [13,19]

**3. Empirical model** 

The modified Maxwell Garnett mixing rule is used
