Calculation of Frequency-Dependent Effective Roughness Dielectric Parameters for Copper Foil Using Equivalent Capacitance Models

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(7)

If the imaginary part is represented through the equivalent conductivity, the corresponding equivalent conductivity is,

(8)

This conductivity will not be high, because it is coming from a lossy effective roughness dielectric in the dielectric phase. Its value is on the order of 10^-2 S, which is similar to a comparatively lossy dielectric.

However, in Region II, the conductivity increases exponentially towards smooth copper level until it reaches the conductivity of the pure copper used on a PCB. Therefore,

(9)

where K₂ is the exponent parameter for conductivity after percolation, and it can be solved from the equation, when σ_p reaches the level at the beginning of percolation, e.g., σ_{p =} 0.01σ_Cu. Percolation threshold is assumed to be 25% of volume concentration of metallic inclusions in the epoxy-resin fiber-filled dielectric matrix [23]. As a reminder, T is the entire thickness of the ERD layer.

Then the conductivity profile function with respect to the coordinate z will be,

(10)

The dielectric profile function in the second conducting layer will be defined as,

(11)

The effective permittivity of the two lossy dielectric layers is calculated through the equivalent capacitor containing two capacitors in series. Both capacitors have gradient fillers. The filler of the first layer is in the non-conducting dielectric phase, and the other is close to percolation, i.e., conducting phase.

(12)

From (12), separating real and imaginary parts, the following ERD parameters can be calculated: DK_r = ε_eff’ and DF_r= tan δ_eff = ε_eff’'/ε_eff’.

Metal Inclusion Profiles in Different Foils

Cross-sectional microscopic (SEM or optical) analysis is used to characterize roughness profile of the foil. For this purpose, typically a signal trace is cut perpendicular to the direction of the electromagnetic wave propagation. The procedure of image processing is described in detail in [17,18,19]. An example of a binary (black-and-white) image of the trace cross-section of VLP foil on PPO Blend substrate is shown in Figure 1. The bottom ("foil", or "matte") side of this foil is rougher that the top ("oxide", or "drum") side.

Figure 1: Binary image of the cross-section of the signal trace of black oxide VLP foil on PPO blend substrate.

The surface roughness profile can be extracted and then quantified using digital image processing based on the analysis of pixels [16]. The average peak-to-valley magnitude of the roughness profile corresponding to the bottom of Figure 1 is shown in Figure 2.

Figure 2: An example of foil roughness profile extracted from the bottom side of the binary image.

Foil surface roughness has the stochastic nature, therefore, along with peak-to-valley values, it can be characterized it in terms of the probability density function (PDF) and autocorrelation function (ACR). Corresponding PDF and ACR curves for VLP foil type are presented in Figure 3. The PDF shows that copper foil surface roughness has normal (Gaussian) distribution, and from ACR it is clearly seen that the roughness is uncorrelated and does not contain any periodicity.

Figure 3: Probability density function histogram (a) and autocorrelation function (b) corresponding to roughness profile in Figure 2.

In many cases (though not always), surface roughness is isotropic, i.e., the PDF is invariant with respect to any direction of the wave propagation. Since the parameters of PDF can be obtained from the profile, the roughness 3D profile can be reconstructed for the future investigation using, for example, Gaussian filter, or any other low-pass filter widely used in digital image processing. The parameters of this filter should be adjusted to get the best correlation with the measured roughness profile [24].

Figure 4: 3D roughness profile surface generated using PDF and Gaussian filter.

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