# Broadband RF characterization and modeling of superconducting coplanar waveguides

Corentin Beilvert, Norman Vivien, Candice Thomas, Jean Charbonnier, Edouard Deschaseaux, Franck Badets, Jean-Philippe Michel

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Corentin Beilvert, Norman Vivien, Candice Thomas, Jean Charbonnier, Edouard Deschaseaux, et al.. Broadband RF characterization and modeling of superconducting coplanar waveguides. IEEE Transactions on Applied Superconductivity, 2024, 35 (3), pp.1-5. 10.1109/TASC.2024.3516210. hal-04965039

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# Broadband RF Characterization and Modeling of Superconducting Coplanar Waveguides

Coretin Beilvert, Norman Vivien, Candice Thomas, Jean Charbonnier, Senior Member, IEEE, Edouard Deschaseaux, Franck Badets, Jean-Philippe Michel

***Abstract***—We present the RF characterization of 2 µm wide superconducting coplanar waveguides (CPW) made of 100 nm thick niobium (Nb) or niobium nitride (NbN) films fabricated on 200 mm silicon wafers. Measurements of the two-ports S-parameters up to 25 GHz in a 4.7-14 K temperature range with an added direct current up to 115 mA were done using a cryogenic probe station. With proper RF calibration and de-embedding procedure, both temperature and DC current dependence of the RF superconducting properties are analyzed. We report a kinetic inductance of 0.15 pH/□, a critical current density of 50 MA/cm² and a critical temperature of 8.9 K for Nb as well as 4.27 pH/□, 3 MA/cm² and 13.8 K for NbN. A telegraphist transmission line model taking into account the BCS theory has been developed and the comparison with the measurements is discussed.

***Index Terms***—two-ports measurement, coplanar waveguide, kinetic inductance, critical current, de-embedding, superconductors

<figure><img src="image_2.png" /><figcaption>Fig. 1. SEM cross section of the routing levels and vias of the interposer.</figcaption></figure>

## I. INTRODUCTION

THE advent of quantum technologies in the recent years have spurred the development of low temperature electronics for the control electronics, the readout circuits or the cooling system itself [1–3]. Superconductors, thanks to their unconventional physical properties such as zero DC resistance and diamagnetism, have benefited from this renewal to find many applications in the fields aforementioned [4, 5]. In particular, superconducting materials can be very efficient for interconnects, linking different components with no losses while thermally isolating them. Superconductor integration can also be very interesting for resonators to help achieving unloaded quality factors exceeding $10^4$ [6]. Those components have to be characterized to obtain sufficiently accurate models enabling the design of optimized systems incorporating them. The case of cryogenic devices is complex to tackle: to reach cryogenic temperatures, the device under test (DUT) has to be placed in a refrigeration system, often a dilution fridge. This implies that the sample generally has to be wire-bonded, and that the signal has to go through a temperature difference of ~300 K and possibly multiple attenuators/amplifiers, which restrict both the bandwidth and the power that can go through the sample but also the calibration process.

Some experiments [7] use the resonant frequency of the device to extract the kinetic inductance, but it restricts the bandwidth to only one frequency and generally requires the addition of a capacitor to lower the resonance. Other groups tried to measure devices on a large bandwidth but as stated in

[8], the calibration is very difficult to achieve. A solution to this issue can be a switching mechanism in order to calibrate directly inside the cryostat [9, 10]. Other authors use probe stations [11, 12], while they generally can't go down to millikelvin temperatures, they are more practical calibration-wise and do not require bonding. This is the type of setup we chose for our experiments.

We present the broadband RF characterization of superconducting coplanar waveguides (CPW) made of Nb or NbN. Using a CPX1 Lakeshore cryoprober, we are able to measure our DUTs without any attenuator or amplifier down to 4.7 K. The use of RF probes allows to use calibration standards as well as measuring the DUTs in the same cool-down up to 25 GHz with an added direct current. The extraction of the measured S-parameters enables obtaining the characteristic impedance $Z_c$, the critical current $I_c$, the critical temperature $T_c$, the inductance $L$ and the kinetic inductance $L_{kin}$ of the device and associated materials. This paper is organized as follows: the fabrication process and the design of the devices are described in II, the measurement setup and de-embedding procedure are presented in III, the model is discussed in IV, the measurements are analyzed in V and the conclusion is drawn in VI.

## II. DESIGN & FABRICATION

This work uses devices embedded within a superconducting routing platform [13] designed for the interconnection of qubit chips with cryo-CMOS control electronics ones. The module is made of two superconducting levels in Nb or NbN connected though TiN vias. The use of superconducting routing is expected to limit signal losses between the interconnected circuits and to thermally isolate them. It also allows the integration of on-chip passive devices with quality factors orders
<figure><img src="image_3.png" /><figcaption>Fig. 2. (a) Photograph of the measurement setup. (b) Simplified diagram of the measurement setup. (c) Microphotograph of a coplanar waveguide.</figcaption></figure>

of magnitude higher than using classic metallic materials. A scanning electron microscopy (SEM) cross section is visible on Fig. 1.

The routing levels are fabricated on top of thermally oxidized Si substrates. Nb and NbN are deposited using physical vapor deposition techniques at 400 °C. Both materials are encapsulated in between a 10 nm-thick adhesion layer of Ti and a 20 nm-thick TiN capping layer, protecting the superconducting layers from the following process steps and eventually from oxidizing. Patterning of these levels is ensured through standard photolithography and dry etch techniques such as in [13]. The devices under study are CPWs, such as the one presented in Fig. 2(c). It is made of a 2 µm wide (W) and 1000 µm long (l) signal line surrounded by two ground planes 3 µm apart from the line. Two GSG (Ground Signal Ground) RF pads are present at each end of the device to provide two-port measurement of the device. These CPWs are made either on the first routing layer (MET1) or the second one (MET2). We also use 500 µm long CPWs for the de-embedding of the 1000 µm line.

### III. SETUP & DE-EMBEDDING PROCEDURE

The cryogenic measurement setup is shown in figure Fig. 2 (a, b). We use a Lakeshore CPX1 cryoprober to cool the DUT down to 4.7 K. There is a slight difference of temperature between the temperature indicated by the prober (4.2 K) and the DUT one due to the sensor not being directly on the device and the thermalization not being perfect. This is accounted for by proper measurements in a PPMS to model the phenomenon, hence the 4.7 K. The RF measurements are performed with an Agilent 5222A Vector Network Analyzer (VNA) from 10 MHz to 25 GHz and a Keithley 2450 DC current generator from 0 to 115 mA. Both instruments are linked with bias Tees so that one flexible RF cable per port descends into the prober to the 150 µm pitch RF GSG probe. The VNA is parameterized with a power of -40 dBm to limit the impact of the measurement on the properties of the materials. It is also short-open-load-through (SOLT) calibrated at the tip of the probes with a CS-5 calibration substrate, which has proven to be suitable

<figure><img src="image_4.png" /><figcaption>Fig. 3. Typical magnitude of measured S-parameters (Nb CPW at 4.7 K).</figcaption></figure>

at cryogenic temperatures [12]. With this setup, S-parameters measurements typically look like Fig. 3.

We use the Y-Cancel de-embedding technique [14] to characterize and remove the influence of the RF pads. This method consists of two steps: first an H matrix is calculated by multiplying the transmission matrix of the whole device by the inverse matrix of the dummy, then the de-embedded DUT is obtained by averaging the left-right mirrored and un-mirrored Y-parameters of H. This means that this method only applies to symmetric 2-port devices. It also has the advantage of using shorter lines as dummies for longer ones, thus optimizing space on the wafer (the resulting de-embedded DUT is the original device minus the pads and the line length of the dummy). Using this method, we find that each pad adds 15 fF of parallel capacitance and ~55 pH of inductance to the measurement. The pads represent 2% of the total number of square of the device, their kinetic inductance can be neglected without de-embedding. It is important to note that there is some variability in the measured inductance of the pads because of the position of the probes on the pads. Leveraging this information, it is possible to either measure two devices and de-embed one with the other or directly take one device and remove the added capacitance and inductance.
The former process is long but mostly removes the influence of the pads, including the kinetic inductance and some of the variability of the position of the probes. The latter speeds up the measurement process while maintaining a sufficient accuracy. We use both techniques, the de-embedding as a failsafe and the corrected to add measurements. It will also be useful for MET1 devices as there were no dummy on this level.

Parameter extraction for CPW is straightforward. The measured two-ports S-parameters can be transformed to characteristic impedance and propagation constant using the telegraphist equations [15].

## IV. MODEL

The total inductance is expressed as follows:

$$L = L_{geom} + L_{kin/□} \frac{l}{W} \quad (1)$$

with $L_{geom}$ the geometric part of the inductance computed via standard conformal mapping based techniques such as in [16]. This type of expression is perfectly convenient for the extractions relative to the NbN, however, the Nb having a weak kinetic inductance (~15% of the geometric for this device) more accurate techniques have to be used such as Finite Elements Methods (FEM) softwares. For this CPW ($W=2$ µm, $S=3$ µm, $l=1000$ µm) the geometric inductance is estimated at 506 nH/m within 5% of error. $L_{kin/□}$ is the kinetic inductance per square, which is dependent on frequency, temperature and current. It can be defined as:

$$L_{kin/□} = \mu_0 \lambda \coth \left( \frac{d}{\lambda} \right) \quad (2)$$

where $\mu_0$ the vacuum permeability, $d$ the thickness of the trace and $\lambda$ the penetration depth which is given by the following expression:

$$\lambda(\omega, T) = \frac{1}{\sqrt{\mu_0 \omega \sigma_2(\omega, T)}} \quad (3)$$

with $\omega$ the angular frequency and $\sigma_2$ the imaginary part of the Mattis-Bardeen complex conductivity [17] defined as:

$$\frac{\sigma_2}{\sigma_N} = \frac{1}{\hbar\omega} \int_{\max(\Delta-\hbar\omega, -\Delta)}^{\Delta} [1-2f(E+\hbar\omega)]g(E)dE \quad (4)$$

$$g(E) = \frac{E(E + \hbar\omega) + \Delta^2}{\sqrt{(E + \hbar\omega)^2 - \Delta^2 \sqrt{\Delta^2 - E^2}}} \quad (5)$$

$\sigma_N$ is the normal state conductivity, $\hbar$ the reduced Planck constant, $\Delta$ the energy gap and $f$ the Fermi-Dirac distribution function. To add the current dependence we used the expression developed by Maki [18] :

$$\frac{L_{kin}(I)}{L_{kin}(0)} = \frac{\Delta_0}{\Delta} \quad (6)$$

where $L_{kin}(0)$ the kinetic inductance at 0A, $\Delta_0$ the gap at 0 A. The gap $\Delta$ can be computed for $(I, T)$ by solving the equations given in [18].

The expression for the critical current, used both in those equations and for our measurements, is derived from

Ginzburg-Landau (GL) theory:

$$I_c(T) = I_{c0}(1 - t^2)^{\frac{3}{2}}(1 + t^2)^{\frac{1}{2}} \quad \text{with } t = \frac{T}{T_c} \quad (7)$$

## V. MEASUREMENTS

### A. Characteristic Impedance

Fig. 4 describes the characteristic impedance $Z_c$ of CPW integrated on MET2 using either Nb or NbN on the frequency band 100 MHz-2 GHz in the 0.33-0.85 $T_c$ temperature range. The plot shows a flat $Z_c$ on the bandwidth for both materials and for all temperatures, which allows us to extract the inductance values at these frequencies. Another element to highlight is the stability of $Z_c$ for the Nb despite the temperature change due to its weak kinetic inductance. In contrast the NbN characteristic impedance is strongly affected by the temperature but benefits from a major gain for the same geometry due to the kinetic inductance. Analysis of the inductance and capacitance shows that it is indeed the inductance change which impacts $Z_c$ and not the capacitance.

<figure><img src="image_5.png" /><figcaption>Fig. 4. Characteristic impedance of the CPW in the 100 MHz - 2 GHz frequency range and 0.33-0.85 $T_c$ temperature range for Nb and NbN.</figcaption></figure>

### B. Kinetic inductance vs. Temperature

By removing the geometric inductance previously calculated, we now plot the kinetic inductance per square of the material with respect to temperature in Fig. 5. We observe a very low kinetic inductance for Nb. The difference between MET1 and MET2 could be explained by the fact that the first level MET1 experiences a larger thermal budget during the fabrication process combined with the high sensitivity of Nb to oxidation. We measured a $T_c$ of 8.9 K and an extrapolated kinetic inductance of 0.15 pH/□ at 0 K for MET2. The small discrepancy at low temperatures is due to the placement of the probes along the different measurements. The material seems to follow pretty well the BCS behaviour, unlike the NbN which on top of having 40 times more kinetic inductance, does not. The main hypothesis is an unbalanced growth of the NbN on the Ti adhesion layer. The material would have gradually
<table><caption>TABLE I
SUMMARY OF EXTRACTED PARAMETERS</caption><thead><tr><th>Material</th><th><i>T</i><sub>c</sub> (K)</th><th><i>L</i><sub>k0</sub> (pH/□)</th><th><i>I</i><sub>c</sub> (mA)</th><th><i>J</i><sub>c</sub> (MA/cm<sup>2</sup>)</th></tr></thead><tbody><tr><td>Nb (MET2)</td><td>8.9</td><td>0.15</td><td>115</td><td>50</td></tr><tr><td>Nb (MET1)</td><td>8.6</td><td>0.20</td><td>-</td><td>-</td></tr><tr><td>NbN</td><td>13.8</td><td>4.27</td><td>6.5</td><td>3</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Kinetic inductance per square as a function of the temperature for MET1/MET2 Nb and NbN. The blue (dashed) line is the model for MET2 (MET1). The red (orange) line is the model for NbN (bilayer NbN).</figcaption></figure>

changing properties along the thickness. We attempt to fit the data with a hybrid model composed of two parallel layers of NbN (MET2a-MET2b) with slightly different properties ($t, T_c, \sigma_N$), shown in orange on Fig. 5. Another possibility could be proximity effects with the adhesion and capping layers. If so, the Nb could also be affected at lower temperatures, lowering the kinetic inductance at 0 K. Finally, properties of NbN seem to be resilient to thermal budget, the extraction being the same for both layers. The extracted values are a $T_c$ of 13.8 K and an extrapolated $L_{k0}$ of 4.27 pH/□. The penetration depth can be extracted as well and we obtain 87 nm for Nb and 447 nm for NbN. All the extractions made in this paragraph seem in good agreement with literature [19] for similar processes and thicknesses.

### C. Critical current & kinetic inductance dependence

To characterize the critical currents of Nb and NbN routing levels, measurements were made using the Keithley 2450 DC current generator. Those measurements can be seen in Fig. 6 (a). The data points are then fitted with equation (7). The extrapolated critical currents at 0 K are 3.6 mA (3 MA/cm²)

for NbN and 115 mA (50 MA/cm²) for Nb. Considering the penetration depth and the dimensions of the layer, the current density of Nb is not homogeneous. Thus, the given current density is an equivalent current density. Additionally, $J_c$ could be improved by using thinner levels or a narrower device line width. Now that $I_{c0}$ is known, the current-dependent inductance model can be plotted against the measurements for the entire temperature range (Fig. 6 (b, c)).

It is noticeable that the kinetic inductance of both materials is very stable under $I_c/2$ for low temperatures. The current-induced variations of the kinetic inductance increase with temperature but the main source of variations is still temperature itself. Table I summarizes the extracted characteristics presented in this study.

## VI. CONCLUSION

In this work, we characterized 2 µm wide superconducting CPW made of 100 nm thick Nb or NbN films fabricated in the 200 mm platform of CEA-LETI. Extracted values are a $T_c$ of 8.9 K, a kinetic inductance at 0 K of 0.15 pH/□, a critical current density of 50 MA/cm² for Nb and a $T_c$ of 13.8 K, a kinetic inductance of 4.27 pH/□ and a critical current density of 3 MA/cm² for NbN. Nb shows great stability over frequency and temperature variation, fits the BCS model, but shows different characteristics depending on the integration level (possibly explained by the thermal budget and the material sensitivity to oxidation). NbN demonstrates a great gain of kinetic inductance and has consistent characteristics whatever the integration level, but does not fit with the BCS model (possibly due to an unbalanced growth on the adhesion layer or due to proximity effects). Those insights and the proposed characterization technique will be of great use for the design of more complex devices such as inductors, adapted transmission lines or filters.

<figure><img src="image_7.png" /><figcaption>Fig. 6. (a) Critical current of Nb (blue) and NbN (red) with respect to the temperature (4.7 K-$T_c$). Kinetic inductance per square of (b) Nb and (c) NbN with respect to current and temperature (model (surface) and measurements (black dots)).</figcaption></figure>
ACKNOWLEDGMENTS

This work is partially supported by the French National
Research Agency under the programme "France 2030" (PEPR
PRESQUILE - ANR-22-PETQ-0002).

A CC-BY 4.0 public copyright license has been applied by
the authors to the present document and will be applied to all
subsequent versions up to the Author Accepted Manuscript
arising from this submission, in accordance with the grant's
open access conditions.

REFERENCES

[1] Y. Guo et al., “Cryogenic cmos rf circuits: A promising approach for large-scale quantum computing,” *IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 71, no. 3, pp. 1619–1625, 2024. DOI: 10.1109/TCSII.2023.3333540.

[2] B. Patra et al., “Cryo-cmos circuits and systems for quantum computing applications,” *IEEE Journal of Solid-State Circuits*, vol. 53, no. 1, pp. 309–321, 2018. DOI: 10.1109/JSSC.2017.2737549.

[3] H. Zu, W. Dai, and A. de Waele, “Development of dilution refrigerators—a review,” *Cryogenics*, vol. 121, p. 103390, 2022. DOI: https://doi.org/10.1016/j.cryogenics.2021.103390.

[4] A. I. Braginski, “Superconductor Electronics: Status and Outlook,” en, *Journal of Superconductivity and Novel Magnetism*, vol. 32, no. 1, pp. 23–44, Jan. 2019. DOI: 10.1007/s10948-018-4884-4.

[5] F. Vignneau et al., *Probing quantum devices with radio-frequency reflectometry*, Issue: arXiv:2202.10516 arXiv:2202.10516 [cond-mat, physics:quant-ph], Feb. 2022. DOI: 10.48550/arXiv.2202.10516.

[6] C. X. Yu et al., “Magnetic field resilient high k- netic inductance superconducting niobium nitride copla- nar waveguide resonators,” *Applied Physics Letters*, vol. 118, no. 5, p. 054001, Feb. 2021. DOI: 10.1063/5. 0039945.

[7] A. J. Annunziata et al., “Tunable superconducting nanoinductors,” en, *Nanotechnology*, vol. 21, no. 44, p. 445202, Nov. 2010. DOI: 10.1088/0957-4484/21/ 44/445202.

[8] A. Alimenti, N. Pompeo, E. Silva, K. Torokhtii, and P. V. García, “Cryogenic characterization of microwave devices,” in *2023 IEEE International Workshop on Technologies for Defense and Security (TechDefense)*, 2023, pp. 262–267.

[9] H. Wang et al., “Cryogenic single-port calibration for superconducting microwave resonator measurements,” vol. 6, no. 3, p. 035015, 2021. DOI: 10.1088/2058-9565/ac070e.

[10] L. Ranzani, L. Spietz, Z. Popovic, and J. Aumentado, “Two-port microwave calibration at millikelvin temperatures,” *Review of Scientific Instruments*, vol. 84, no. 3, p. 034704, Mar. 2013. DOI: 10.1063/1.4794910.

[11] A. S. Boaventura et al., “Microwave Modeling and Characterization of Superconductive Circuits for Quantum Voltage Standard Applications at 4 K,” *IEEE Transactions on Applied Superconductivity*, vol. 30, no. 2, pp. 1–7, Mar. 2020. DOI: 10.1109/TASC.2019.2963403.

[12] B. Patra, M. Mehrpoo, A. Ruffino, F. Sebastiano, E. Charbon, and M. Babaie, “Characterization and analysis of on-chip microwave passive components at cryogenic temperatures,” *IEEE Journal of the Electron Devices Society*, vol. 8, pp. 448–456, 2020. DOI: 10.1109/JEDS. 2020.2986722.

[13] C Thomas et al., “Superconducting routing platform for large-scale integration of quantum technologies,” *Materials for Quantum Technology*, vol. 2, no. 3, p. 035001, 2022. DOI: 10.1088/2633-4356/ac88ae.

[14] A. Mangan, S. Voinigescu, Ming-Ta Yang, and M. Tazlauanu, “De-embedding transmission line measurements for accurate modeling of IC designs,” *IEEE Transactions on Electron Devices*, vol. 53, no. 2, pp. 235–241, Feb. 2006. DOI: 10.1109/TED.2005.861726.

[15] D. M. Pozar, *Microwave Engineering*. 1990.

[16] K. Watanabe, K. Yoshida, and T. A. Kohjiro, “Kinetic inductance of superconducting coplanar waveguides,” *Japanese Journal of Applied Physics*, vol. 33, no. 10R, p. 5708, 1994. DOI: 10.1143/JJAP.33.5708.

[17] D. C. Mattis and J. Bardeen, “Theory of the Anomalous Skin Effect in Normal and Superconducting Metals,” en, *Physical Review*, vol. 111, no. 2, pp. 412–417, Jul. 1958. DOI: 10.1103/PhysRev.111.412.

[18] K. Maki, “The Behavior of Superconducting Thin Films in the Presence of Magnetic Fields and Currents*),” *Progress of Theoretical Physics*, vol. 31, no. 5, pp. 731–741, May 1964.

[19] S. K. Tolpygo et al., “Progress Toward Superconductor Electronics Fabrication Process With Planarized NbN and NbN/Nb Layers,” *IEEE Transactions on Applied Superconductivity*, vol. 33, no. 5, pp. 1–12, Aug. 2023. DOI: 10.1109/TASC.2023.3246430.