# High-Frequency Modeling of Coplanar Waveguides Including Surface Roughness

### Konstantin Lomakin

### Klaus Helmreich

### Uwe Arz

An existing analytical transmission line model to describe propagation
properties of coplanar waveguides including dispersion and radiation effects
was extended to take into account surface roughness of conductor traces. The
influence of parasitics is successively included in the simulation and
compared to measurements. The device under test (DUT) was fabricated on an
Al_{2}O_{3} wafer. A metal and ceramic chuck was utilized during
measurements up to 120 GHz. The extended model is then capable of precisely
predicting propagation properties in a wide frequency range and can now be
used for calibration purposes like the development of uncertainty budgets.

Coplanar waveguides (CPWs) are frequently used in all different kinds of planar circuits like (printed circuit boards) PCBs and on-wafer applications. Especially the latter – ubiquitous nano- and microelectronics – utilize this transmission line type particularly. The exact knowledge of its high-frequency properties thus is essential for modern electronics with increasing data-rates and consequently increasing application frequencies.

Nowadays numerical full-wave electro-magnetic simulators are available that are capable of accounting for parasitic effects like dispersion, radiation and conductor loss effects (Schmückle et al., 2011). Nevertheless analytical transmission line models are required for developing reliable uncertainty budgets for calibration purposes (Arz et al., 2017) and moreover have fundamental importance to microwave design.

Such an analytical model was presented in the early nineties (Heinrich, 1993). This quasi-TEM approach describes the electrical behavior of a CPW from DC to a certain frequency – depending on CPW dimensions – from where on radiation and additional dispersion, due to coupling to higher order modes, take place. These effects were added to the CPW model in (Schnieder et al., 2003), so that modeling of CPW properties at high frequencies is possible now. But at high frequencies the skin depth decreases to the order of surface roughness, so that the assumption of ideal smooth conductor walls does not hold anymore and roughness strongly impacts on transmission properties of any kind of waveguide.

This paper shows the application of a roughness model (Gold and Helmreich, 2012) that is
capable of accounting for impact on loss and delay of transmission lines to
the contemplated CPW model. This new extension finally allows for precisely
describing properties of CPWs including all known parasitic effects. The
model predictions are compared to measurements of CPWs on Al_{2}O_{3} with
both metal and ceramic chuck and show a very good agreement in both
scenarios.

In the following the geometrical parameters *w*_{g}, *w*, *t*, *s* and *h* are
utilized as shown in the sketched cross-section of a CPW in
Fig. 1. Additionally the material parameters relative
permittivity *ε*_{r} of the substrate, dielectric loss tangent
tan *δ* and conductivity *σ* of the traces are necessary.

The regarded transmission lines are purely coplanar, i.e. there is no
additional ground plane beneath the dielectric. However, to rest the DUT
during measurement, a so called *chuck* is used. Depending on its
material, metal or ceramic, there is an impact on the measurement and
consequently has to be considered during simulation, which is done by
assuming an infinite substrate in the case of the ceramic chuck. This
assumption is justified, since the height of the chuck is very large compared
to the substrate, so that the electromagnetic field does not penetrate
through it. Additionally, the relative permittivity of the chuck and
substrate are assumed to be equal. The metal chuck results in a boundary
condition respectively after the specified substrate height. A more detailed
insight of the impact from different chuck permittivities is given in
Phung et al. (2018a, b).

The CPW was fabricated on an Al_{2}O_{3} substrate, which was
characterized separately with a split cylinder resonator (Janezic, 2003). The
relative permittivity in the frequency range from 8 to 22 GHz was
measured as 9.7, the loss tangent about $\mathrm{1.25}\times {\mathrm{10}}^{-\mathrm{4}}$ to
$\mathrm{1.5}\times {\mathrm{10}}^{-\mathrm{4}}$. The nominal dimensions of the lines are *w*=50 µm, *s*=25 µm, *w*_{g}=270 µm. Due to the electroplating
process the conductive traces are subject to a variation of their thickness
across the wafer. A figure of 6.5 µm therefore is used in the
following investigations. Likewise the conductivity *σ*_{DC} is
affected by the electroplating process, since it is always lower compared to
bulk metal. Because of that we use a conductivity value of 36 MS m^{−1}.

Thus, measurements were performed whereby the complex propagation coefficient
*γ* of the CPW is obtained by a multiline calibration method
(Marks, 1991). The measured responses are presented as attenuation *α*=Re(*γ*) and effective permittivity ${\mathit{\epsilon}}_{\mathrm{r},\phantom{\rule{0.125em}{0ex}}\text{eff}}={\left({c}_{\mathrm{0}}/{v}_{\mathrm{p}}\right)}^{\mathrm{2}}$, in which *v*_{p} in turn is the phase
velocity ${v}_{\mathrm{p}}=\mathit{\omega}/\text{Im}\left(\mathit{\gamma}\right)$.

In the next sections, the CPW model with and without radiation and dispersion effects is compared to the measured propagation properties. Then the application of the surface roughness model is explained. For this purpose the relationship between independently measured surface profile data, magnetic field and resulting effective, frequency dependent material parameters and necessary adaptions in the CPW model are shown.

## 3.1 CPW Model without dispersion and radiation

In analytical transmission line models, the electromagnetic properties are
described with per unit length (pul) parameters *L*^{′} and *C*^{′}, the loss
mechanisms with *R*^{′} and *G*^{′} for an infinitesimal short transmission line
segment *d*_{z}. From those parameters, the complex propagation coefficient can
be calculated:

It's worth to note, that both outer and inner inductance contribute to the total pul inductance ${L}^{\prime}={L}_{\mathrm{o}}^{\prime}+{L}_{\mathrm{i}}^{\prime}$. The inner inductance represents the magnetic field penetrating the conductor due to its finite conductivity, i.e. the skin effect.

The quasi-TEM model approach (Heinrich, 1993) includes conductor loss, substrate
loss and finite metallization thickness. With the geometrical and material
properties from section II, attenuation *α* and effective, relative
permittivity *ε*_{r, eff} can be calculated analytically.
These responses are compared to the measurement in Figs. 2
and 3. To illustrate the stepwise improvements in the
following, the deviation of simulation and measurement is plotted as relative
error (gray filled).

The simulation is in good agreement with the measured responses at lower
frequencies. The deviation of simulated and measured attenuation gets
significant above 40 GHz. The effective rel. permittivities in Fig. 3 mainly show an offset, which could be explained by
uncertainties in *ε*_{r} of substrate, metallization thickness *t* or
surface roughness impact. Besides this, the shape of the response clearly
drift apart above 40 GHz.

## 3.2 CPW Model with radiation and dispersion

The CPW model (Schnieder et al., 2003) is an extension to the quasi-TEM model (Heinrich, 1993) now considering radiation and dispersion effects. As a result, it delivers an attenuation response (Fig. 4), that adapts the measured response much better.

Particularly, with regard to the bend towards higher attenuation that is
observed above 60 GHz. This response strongly differs from the one
calculated with the quasi-TEM model having only a $\sqrt{f}$- and
*f*-dependency and the relative error decreases below 10 %. Including
dispersion effects in the model also leads to a simulated effective
permittivity (Fig. 5) which shows a similar response as
the measured one. Especially, the increase of phase delay above 60 GHz
is now depicted. The relative error stays below 3 % and indicates a
merely constant offset over the entire frequency range.

## 3.3 Surface Roughness

Generally a model should have an underlying physical notion, a minimum number
of parameters and of course it must predict measurement results with
reasonable accuracy. Especially a model for the interaction of
electromagnetic fields with rough surfaces should utilize input parameters
for surface properties, that are well known and specified in surface
metrology. For example the RMS^{1}-roughness *R*_{q} is an estimate for the
standard deviation of the surface profile from the mean surface. It is
defined in ISO 4287 (1997-04) among other surface texture parameters.

The Gradient Model presented by Gold and Helmreich (2012) fulfills the stated requirements
and is capable of predicting roughness impact on both loss and delay.
Therefore the rough surface is modeled by a continuous conductivity
transition from the dielectric to the bulk conductivity *σ*_{DC}.
In the following the parameter *x* describes the depth counting positively
into the material. For surface profiles that are distributed normally, it
only requires one additional parameter, the RMS-roughness *R*_{q} to calculate the conductivity profile:

Many surface profile measurement systems like tactile or optical ones not only deliver statistical parameters, but also the bearing area curve (BAC) or Abott-Firestone curve (Abbott and Firestone, 1933), which corresponds to a cumulative probability density function. In cases of not very random surface profiles, such as with chatter marks for example, the Gradient Model is also capable of dealing with the measured BAC directly.

### 3.3.1 Surface Profile Measurements

The mean response from four BAC measurements at the top of the Al_{2}O_{3}
wafer metallization is shown in Fig. 6. The standard deviation is
marked gray in the background. From the mean BAC, *R*_{q} was estimated to be
*R*_{q}=63.5 nm. Then again a CDF can be calculated with Eq. (2),
which is also shown in Fig. 6 and multiplied with
*σ*_{DC} used as conductivity profile to calculate the magnetic
field in the rough surface. It can be clearly seen, that the magnitude of the
magnetic field differs from an exponential function, which is the solution of
classical skin effect for ideally smooth surfaces.

### 3.3.2 Application using effective material parameters

From the magnetic field all other relevant quantities, e.g. loss power
density, magnetic field energy, etc. can be calculated. Since most analytical
transmission line models assume ideally smooth surfaces, the application of
the Gradient Model utilizing effective, frequency dependent material
parameter was shown in Gold and Helmreich (2017). The effective conductivity
*σ*_{eff} represents roughness impact on loss (*R*^{′}) and the
effective permeability *μ*_{r, eff} the impact on inner inductance
(${L}_{\mathrm{i}}^{\prime}$). They are determined by comparing loss power density or magnetic
field energy of the rough and smooth surface respectively. The responses for
*R*_{q}=63.5 nm are shown in Fig. 7. In this case, the
effective material values are different from the ideal case, i.e.
*σ*_{eff}=*σ*_{DC} and ${\mathit{\mu}}_{\mathrm{r},\phantom{\rule{0.125em}{0ex}}\text{eff}}=\mathrm{1}$ but
relatively close to it. That is because even for 100 GHz the skin
depth is roughly four times larger than *R*_{q}. Or in other words the
conductors are very smooth – but not ideally smooth – in the regarded
frequency range.

## 3.4 CPW Model with radiation, dispersion and roughness

The advantage of using effective material parameters to depict roughness impact is, that they can be utilized in any model assuming ideally smooth surfaces. This is done by simply replacing material with effective material parameters, one only has to take care of their frequency dependence.

The CPW model distinguishes three frequency sections: The quasi-static case,
a transition region and the skin effect region. Since the Gradient Model
assumes the skin effect, only skin effect and transition region has to be
adapted. In the latter, the conductivity *σ* is replaced by
*σ*_{eff} in the pul parameter *R*^{′}:

where index “*c*” denotes the signal and “g” the ground conductor traces
respectively. The effective permeability *μ*_{r, eff} is added to pul
inductance *L*^{′} in this region denoted as *L*_{z2}:

In the skin effect region the modifications of the pul resistances ${R}_{c}^{\prime}$, ${R}_{\mathrm{g}}^{\prime}$ are as follows:

The pul inductance *L*^{′} in the skin effect region becomes:

From the original document (Heinrich, 1993) only Eqs. (6),
(7), (9) and expressions for ${R}_{c\mathrm{1}}^{\prime}$, ${R}_{\mathrm{g}\mathrm{1}}^{\prime}$ and ${L}_{z\mathrm{2}}^{\prime}$ were
modified. The variables *F*_{L} and *F*_{0} are geometry factors and are also
derived in (Heinrich, 1993). All other equations can be left untouched.

As a result we obtain a model including most known parasitics, that can precisely predict propagation properties of CPWs. The additional consideration of surface roughness impact leads to an improvement in the frequency range from 50 to 100 GHz regarding the attenuation in Fig. 8 utilizing a metal chuck. The relative error stays below 5 % up to 100 GHz. Above 100 GHz measurement uncertainties become larger due to influences from parasitic modes such as e.g. substrate modes.

The surface roughness impact on phase delay and effective rel. permittivity respectively is typically nearly constant over the regarded frequency range (Fig. 9), so that the simulated and measured responses only lead to an effective error below 0.5 %.

Measurements of the same wafer on a ceramic chuck lead to slightly different
results. Especially, the attenuation does not show a distinctive bend above
60 GHz. Figure 10 shows, that it is much
closer to a $\sqrt{f}$- and *f*-dependency than the responses obtained from
measurements utilizing a metal chuck.

In the simulation the ceramic chuck was considered by assuming an infinite substrate height, since it consists of a similar material.

Like in the case with metal chuck, the relative error only raises to noteworthy figures above 100 GHz.

Also the effective relative permittivity in case of a ceramic chuck is predicted very well, as can be seen in Fig. 11 with a similar relative error as for the measurements on a metal chuck.

An existing CPW model was expanded utilizing frequency dependent, effective
material parameters. The necessary modifications were explained in detail.
Those effective parameters are calculated with a surface roughness model that
considers roughness impact on both phase and delay. Its input parameter, the
RMS-roughness *R*_{q}, was estimated from measured bearing area curves. The
obtained model now includes parasitics like dispersion, radiation and
conductor surface roughness.

The DUT was fabricated on an Al_{2}O_{3} substrate and was measured
utilizing a ceramic and a metal chuck. The simulations with the extended
model precisely predict the measured responses, i.e. attenuation and phase
delay in both cases up to 100 GHz. The impact from dispersion and
surface roughness on attenuation and effective relative permittivity was
shown by successively including those parasitics in simulations.

Precise analytical transmission line models including parasitics are fundamental for developing reliable uncertainty budgets for calibration purposes and also have their importance in design.

The data are available from the authors upon request.

The authors declare that they have no conflict of interest.

The authors are grateful to Rohde & Schwarz for manufacturing the calibration substrate, to Thorsten Probst from PTB for performing the on-wafer measurements, and to Dylan Williams from NIST for providing the initial layout of the substrate.

Furthermore, the authors would like to thank Franz-Josef Schmückle from Ferdinand-Braun-Institute for supplying program code on radiation and dispersion effects of a CPW.

The authors acknowledge support by the European Metrology Programme for Innovation and Research (EMPIR) Project 14IND02 “Microwave measurements for planar circuits and components”. The EMPIR program is co-financed by the participating countries and from the European Union's Horizon 2020 research and innovation program.

This paper was edited by Thorsten Schrader and reviewed by two anonymous referees.

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root mean square.