**2. Methodology**

Interconnects often form regular gratings (e.g. bus structures), hence an infinite metallic grating in a homogenous dielectric host, as shown in Fig. 1., represents a straightforward but broadly applicable model.

**Figure 1.** Diagram of 2-D grating structure studied here. The dashed line represents the boundary of the homogenised equivalent (not shown).

Wide parameter space for the dimensions is considered, as follows. The interconnect pattern density is initially limited by the design rules within 20% - 80% metal fill [14]. A statistical analysis of the pattern density of a real chip showed that, typically, the maximum pattern density in actual metal layers does not exceed 60% [15]. Thus metal fill factors *f* in the range 0.3 - 0.6 is considered. While metal layer height is fixed for any given layer in any given process, track width is less restricted. Aspect ratios (xAR) are continuing to increase as technology develops [16], hence structures with narrowed down [4,5] values within 1.4 ≤ xAR ≤ 3 are studied. Due to the growing use of low-k dielectrics a host materials with permittivity e in the range 1 ≤ <sup>e</sup> ≤ 11.7 are considered. The interconnect pitch is often measured in micrometres or nanometres, whereas the wavelength λ of the clock signal is typically measured in centimetres. Thus, we can expect to successfully apply an appropriate effective medium approximation because the condition << λ/4 is met [17]. Here, the 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 and all but the top layer of interconnects.

The modified Maxwell Garnett mixing rule is used

200 Dielectric Material

scaling factor has more rapid increment [4, 5].

**2. Methodology** 

but broadly applicable model.

the homogenised equivalent (not shown).

Λ

*<sup>f</sup>*<sup>Λ</sup>

incident plane wave

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

Interconnects often form regular gratings (e.g. bus structures), hence an infinite metallic grating in a homogenous dielectric host, as shown in Fig. 1., represents a straightforward

**E**TM

**Figure 1.** Diagram of 2-D grating structure studied here. The dashed line represents the boundary of

Wide parameter space for the dimensions is considered, as follows. The interconnect pattern density is initially limited by the design rules within 20% - 80% metal fill [14]. A statistical analysis of the pattern density of a real chip showed that, typically, the maximum pattern density in actual metal layers does not exceed 60% [15]. Thus metal fill factors *f* in the range 0.3 - 0.6 is considered. While metal layer height is fixed for any given layer in any given process, track width is less restricted. Aspect ratios (xAR) are continuing to increase as technology develops [16], hence structures with narrowed down [4,5] values within 1.4 ≤ xAR ≤ 3 are studied. Due to the growing use of low-k dielectrics a host materials with permittivity e in the range 1 ≤ <sup>e</sup> ≤ 11.7 are considered. The interconnect pitch is often measured in micrometres or nanometres, whereas the wavelength λ of the clock signal is typically measured in centimetres. Thus, we can expect to successfully apply an appropriate effective medium approximation because the condition << λ/4 is met [17]. Here, the

**<sup>K</sup>** dielectric

h

metal

$$\mathfrak{E}\_{eff} = \mathfrak{E}\_e + \Psi f \mathfrak{E}\_e \frac{\mathfrak{E}\_i - \mathfrak{E}\_e}{\mathfrak{E}\_i + \mathfrak{D}\mathfrak{E}\_e - f\left(\mathfrak{E}\_i - \mathfrak{E}\_e\right)} \tag{1}$$

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 Drude model [13,19]

$$\left(\varepsilon\_i(o) = 1 - \frac{o\nu\_p^2}{o(o+j\gamma)}\right) \tag{2}$$

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 the analytical and numerical calculations.

#### **3. Empirical model**

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

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

$$
\Psi(\mathbf{x}\_{AR}) = \mathbf{a} \cdot \mathbf{x}\_{AR} + \mathcal{J} \tag{3}
$$

Empirical Mixing Model for the Electromagnetic Compatibility Analysis of On-Chip Interconnects 203

**Figure 2.** Plot of the scaling factor obtained for an example grating structure. The fitted values (crosses) show good agreement with approximated data (lines). Grating parameters: *f* = 0.5, = 100 μm,

**Figure 3.** Plot of the reflection coefficient obtained from a subset of the gratings studied, as a function of aspect ratio and host permittivity. Results from the homogenised model are drawn as lines, while those from the detailed structure simulated with RCWA are plotted as markers. Fixed parameters: ν = 5 GHz,

ν = 5 GHz, = 6.25, 1.4 ≤ xAR ≤ 3.

*f* = 0.5.

where the coefficients *α, β* are determined by linear regression from data obtained from nearly 5000 simulations spanning a four dimensional parameter space. The coefficients are represented as a linear function of metal fill factor by

$$k(f) = k\_1 \cdot f + k\_{2'} \qquad k = \{\alpha, \beta\} \tag{4}$$

where k1, k2 are well approximated by

$$\begin{aligned} k\_1(\nu) &= k\_{11} \cdot \nu + k\_{12}, \\ k\_2(\nu) &= k\_{21} \cdot \nu + k\_{22}, \qquad k = \{\alpha, \beta\} \end{aligned} \tag{5}$$

where ν is the frequency (in units of GHz), and factors k11, k12, k21, k22 are presented in Table 1.


**Table 1.** Coefficients for calculation of the scaling factor .

The fit of the model was assessed using a linear least square method. Figure 2 illustrates the good agreement between obtained from a 'brute force' fitting algorithm and that from our linear approximation for an example grating structure. The grating has *f* = 0.5, and the illumination frequency is 5 GHz.

In Fig. 3 the reflection coefficient obtained using RCWA and homogenised model for a sample of 27 different structures are depicted. For frequencies in the range 1 GHz ≤ ν ≤ 10 GHz the error between RCWA results for the detailed structure and those obtained for the homogenised structure is less than 2.5% for reflection coefficient and 0.2% for the transmission coefficient (not shown here) when 0.3 ≤ *f* ≤ 0.6 and θ ≤ ±30. The accuracy of the model for reflection coefficient calculations tends to improve with an increase of the metal fill factor. For structures with metal fills 0.4 ≤ *f* ≤ 0.6 the error varies between 0 – 1.5%. When the model is applied, without modification, to interconnects with a trapezoidal cross section, sometimes found in fabricated structures, the error remains similar as for the empirical model presented in [4,5] and is below 5% for sidewalls with angles of up to 5 and incident angle up to 30.

The calculated effective permittivity varies according to the particular mixing rules used to analyse a given mixture. However, there are theoretical bounds to the range of calculated effective permittivieties. For the compound of the two dielectrics the effective permittivity calculated from the Maxwell-Garnett mixing rule has to fall in between the following bounds [13]

202 Dielectric Material

Table 1.

bounds [13]

() *AR AR x x*

represented as a linear function of metal fill factor by

**Table 1.** Coefficients for calculation of the scaling factor .

angles of up to 5 and incident angle up to 30.

illumination frequency is 5 GHz.

where k1, k2 are well approximated by

where the coefficients *α, β* are determined by linear regression from data obtained from nearly 5000 simulations spanning a four dimensional parameter space. The coefficients are

1 2 *kf k f k k* () , {,}

( ) , {,}

where ν is the frequency (in units of GHz), and factors k11, k12, k21, k22 are presented in

α11 0.0064 β11 -0.0341 α12 0.2309 β12 -1.7494 α21 -0.0037 β21 0.0213 α22 -0.2346 β22 2.8473

The fit of the model was assessed using a linear least square method. Figure 2 illustrates the good agreement between obtained from a 'brute force' fitting algorithm and that from our linear approximation for an example grating structure. The grating has *f* = 0.5, and the

In Fig. 3 the reflection coefficient obtained using RCWA and homogenised model for a sample of 27 different structures are depicted. For frequencies in the range 1 GHz ≤ ν ≤ 10 GHz the error between RCWA results for the detailed structure and those obtained for the homogenised structure is less than 2.5% for reflection coefficient and 0.2% for the transmission coefficient (not shown here) when 0.3 ≤ *f* ≤ 0.6 and θ ≤ ±30. The accuracy of the model for reflection coefficient calculations tends to improve with an increase of the metal fill factor. For structures with metal fills 0.4 ≤ *f* ≤ 0.6 the error varies between 0 – 1.5%. When the model is applied, without modification, to interconnects with a trapezoidal cross section, sometimes found in fabricated structures, the error remains similar as for the empirical model presented in [4,5] and is below 5% for sidewalls with

The calculated effective permittivity varies according to the particular mixing rules used to analyse a given mixture. However, there are theoretical bounds to the range of calculated effective permittivieties. For the compound of the two dielectrics the effective permittivity calculated from the Maxwell-Garnett mixing rule has to fall in between the following

1 11 12 2 21 22 ( ) ,

*kkk k* 

 

*kk k*

  

> 

(3)

(4)

(5)

**Figure 2.** Plot of the scaling factor obtained for an example grating structure. The fitted values (crosses) show good agreement with approximated data (lines). Grating parameters: *f* = 0.5, = 100 μm, ν = 5 GHz, = 6.25, 1.4 ≤ xAR ≤ 3.

**Figure 3.** Plot of the reflection coefficient obtained from a subset of the gratings studied, as a function of aspect ratio and host permittivity. Results from the homogenised model are drawn as lines, while those from the detailed structure simulated with RCWA are plotted as markers. Fixed parameters: ν = 5 GHz, *f* = 0.5.

$$\mathcal{E}\_{\text{eff},\text{max}} = f \mathcal{E}\_i + (1 - f)\mathcal{E}\_{e'}.$$

$$\mathcal{E}\_{\text{eff},\text{min}} = \frac{\mathcal{E}\_i \mathcal{E}\_e}{f \mathcal{E}\_e + (1 - f)\mathcal{E}\_i}.$$

Empirical Mixing Model for the Electromagnetic Compatibility Analysis of On-Chip Interconnects 205

Experimental validation was carried out by the free space measurement of S-parameters of an air-copper grating structure (Λ = 500μm, *f* = 0.3, AR = 1) attached to a Rogers 4350 dielectric plate (thickness 762μm, r = 3.66) illuminated by a plain wave. A pair of horn antennas operated at the X-band (8.2 – 12.4 GHz) frequencies with Teflon's hemispherical lenses connected to the network analyzer was used. The plane wave illumination focused on a relatively small area was achieved by the special equipment arrangement. A free space calibration method along with smoothing procedure was implemented in order to eliminate systematic errors occurring in the measurement data [22,23]. A 2-D finite difference time domain (FDTD) was defined as shown in Fig. 5, with the case of detailed grating structure in

**Figure 4.** Plot of the real part of the effective refractive index of the grating structure compared with

The domain size was 60 cells in x by 1 cell in y and 8203 cells in z direction. The space increment in both directions was set to 5μm and it was ensured that the domain size in the z-direction was at least a half wavelength from each of the absorbing Perfect Matched Layer (PML) boundaries as the behaviour of these boundaries is not reliable in the presence of evanescent fields. The grating structure is periodic in x-direction, with one period of the

theoretical bounds. Grating parameters: xAR = 1, = 11.7, = 100 μm, ν = 5 GHz, 0.2 ≤ *f* ≤ 0.6.

**4. Experimental validation** 

Fig. 5(a) and its homogenised equivalent in Fig. 5(b).

These bounds are also called Wiener bounds. The upper limit for the effective permittivity εeff,max is defined for a layered material with boundaries between inclusions and host dielectric parallel to the field vector. The lower bound εeff,min is obtained for the case where the field vector is perpendicular to the boundaries between inclusions and host. Since the Wiener bounds are defined for anisotropic mixtures, stricter bounds, Hashin-Shtrikman bounds, have been defined for the statistically homogenous, isotropic and three dimensional mixtures. The upper and the lower bounds are as follows

$$\begin{aligned} \mathcal{E}\_{eff,\text{max}} &= \mathcal{E}\_i + \frac{1 - f}{\frac{1}{\mathcal{E}\_\varepsilon - \varepsilon\_i} + \frac{f}{3\varepsilon\_i}}, \\ \mathcal{E}\_{eff,\text{min}} &= \mathcal{E}\_\varepsilon + \frac{f}{\frac{1}{\mathcal{E}\_i - \varepsilon\_\varepsilon} + \frac{1 - f}{3\varepsilon\_\varepsilon}}, \end{aligned}$$

where it is assumed that εi > εe. The lower limit corresponds to the Maxwell-Garnett mixing rule whereas the upper limit is the Maxwell-Garnett rule for the complementary mixture obtained by transferring the constituents: εi → εe, ε<sup>e</sup> → εi, *f* → 1 - *f*.

It was verified that in a set of about 5000 simulations of the grating structure run to define the empirical model all effective refractive indices (neff = √εeff) are well within the Wiener bounds. Nevertheless, the predicted neff has values close to the lower limit. This is related to the specific alignment of the grating structure (single layer of interconnects) and the angle of incidence wave. Such regular and linearly distributed arrangement of the inclusions with the field vector perpendicular to the grating surface results in an effective permittvity from the bottom range of the possible values defined by Wiener bounds. The upper limit is several orders higher in magnitude, hence even if satisfied, for the purpose of the analyses of this particular grating structure it can be lowered by replacing it with the Hashin-Shtrikman lower limit. It is illustrated in Fig. 4, for a random structure, that the real parts of neff obtained from the empirical model are within the lower limits of the Wiener and Hashin-Shtrikman bounds.

The more strict Hashin-Shtrikman bounds overestimate the obtained values of neff. Hence these limits are based on the Maxwell-Garnett mixing rule for the complementary mixtures and the lower limit is just the classical Maxwell-Garnett rule with εi > εe. Therefore, for the analysed interconnect grating structure it can be assumed that the upper bound for the effective refractive index is the classical Maxwell-Garnett rule whereas the lower bound is the Wiener lower limit.
