**4. Experimental validation**

204 Dielectric Material

Shtrikman bounds.

the Wiener lower limit.

,max

*eff i e*

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

(1 ) ,

 

. (1 )

<sup>1</sup> , <sup>1</sup>

*f f*

*ei i*

 

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

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-

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

*f*

*ie e*

3

 

> 

, <sup>1</sup> <sup>1</sup> 3

*f*

*i e*

*f f*

*f f*

 

*e i*

,min

*eff*

,max

obtained by transferring the constituents: εi → εe, ε<sup>e</sup> → εi, *f* → 1 - *f*.

*eff i*

 

 

,min

*eff e*

mixtures. The upper and the lower bounds are as follows

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

**Figure 4.** Plot of the real part of the effective refractive index of the grating structure compared with theoretical bounds. Grating parameters: xAR = 1, = 11.7, = 100 μm, ν = 5 GHz, 0.2 ≤ *f* ≤ 0.6.

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 grating defined in the domain. The structure was illuminated by a wave with frequencies within 8 – 12 GHz in steps of 1 GHz applied at the top of the domain and propagated in the negative z-direction.

Empirical Mixing Model for the Electromagnetic Compatibility Analysis of On-Chip Interconnects 207

**Figure 6.** Plot of the reflection coefficient for a grating structure. The experimental data follows the same trend as the numerical FDTD and FEM calculations obtained for grating, homogenised and reference structure. Grating parameters: *f* = 0.3, = 500 μm, AR = 1; substrate: thickness 762μm, r = 3.66.

**Figure 7.** Plot of the transmission coefficient for a grating structure. The experimental data follows the same trend as the numerical FDTD and FEM calculations obtained for grating, homogenised and reference structure. Grating parameters: *f* = 0.3, = 500 μm, AR = 1; substrate: thickness 762μm, r = 3.66.

The homogenised equivalent structure was obtained by replacing the grating layer with a solid dielectric. The dielectric properties were calculated from the modified Maxwell-Garnett mixing rule. The value of scaling factor was empirically found as due to the structural difference between experimental settings and the structure studied in order to define the empirical model, the straightforward application of the empirical model underestimated factor . It was verified that , when equal to 2.5, gives good approximation of the calculated effective permittivity eff.

**Figure 5.** Diagram of the 2-D FDTD simulation domain. The domain is one period wide and periodic in (a) x-direction, with its (b) homogenised equivalent

The FDTD and analytical calculations, based on characteristic matrix method, of reflection and transmission coefficients for the gratings with structural period 500μm and its homogenised equivalent are plotted in Fig. 6 and Fig. 7 along with measured return and transmission losses respectively. In order to validate the proposed approach for a wider range of frequencies, 1 - 18 GHz, numerical calculations were performed using finite element method (FEM) [24]. This analysis shows that the results of the two numerical techniques and measured results follow the same trend over a wide range of frequencies and allowed extrapolation of the measured reflection and transmission coefficients outwith the measured domain. Simulated and measured results are in good qualitative agreement and the observed deviation tends to increase simultaneously with frequency increase.

206 Dielectric Material

negative z-direction.

Periodic boundary

*x*

*K, E*

*TM*

*z*

*y*

*E*

*TE*

Direction of Propagation

grating defined in the domain. The structure was illuminated by a wave with frequencies within 8 – 12 GHz in steps of 1 GHz applied at the top of the domain and propagated in the

The homogenised equivalent structure was obtained by replacing the grating layer with a solid dielectric. The dielectric properties were calculated from the modified Maxwell-Garnett mixing rule. The value of scaling factor was empirically found as due to the structural difference between experimental settings and the structure studied in order to define the empirical model, the straightforward application of the empirical model underestimated factor . It was verified that , when equal to 2.5, gives good

> Periodic boundary

> > 0.150 mm

Periodic boundary

/2

**n** Rogers dielectric

air

PML

air

PML

Source

0.762 mm

/2

**Figure 5.** Diagram of the 2-D FDTD simulation domain. The domain is one period wide and periodic in

*(a) (b)*

The FDTD and analytical calculations, based on characteristic matrix method, of reflection and transmission coefficients for the gratings with structural period 500μm and its homogenised equivalent are plotted in Fig. 6 and Fig. 7 along with measured return and transmission losses respectively. In order to validate the proposed approach for a wider range of frequencies, 1 - 18 GHz, numerical calculations were performed using finite element method (FEM) [24]. This analysis shows that the results of the two numerical techniques and measured results follow the same trend over a wide range of frequencies and allowed extrapolation of the measured reflection and transmission coefficients outwith the measured domain. Simulated and measured results are in good qualitative agreement and the observed deviation tends to increase simultaneously with frequency

approximation of the calculated effective permittivity eff.

(a) x-direction, with its (b) homogenised equivalent

**PEC**

Source

Rogers dielectric

air

PML

air

PML

increase.

**Figure 6.** Plot of the reflection coefficient for a grating structure. The experimental data follows the same trend as the numerical FDTD and FEM calculations obtained for grating, homogenised and reference structure. Grating parameters: *f* = 0.3, = 500 μm, AR = 1; substrate: thickness 762μm, r = 3.66.

**Figure 7.** Plot of the transmission coefficient for a grating structure. The experimental data follows the same trend as the numerical FDTD and FEM calculations obtained for grating, homogenised and reference structure. Grating parameters: *f* = 0.3, = 500 μm, AR = 1; substrate: thickness 762μm, r = 3.66.
