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IEEE Transactions on Microwave Theory and Techniques

Cryogenic On-Chip In Situ S -Parameter Calibration Using Superconducting Coplanar Waveguides

Jeremy N. Thomas, Johannes Paul Hoffmann, Nathan E. Flowers-Jacobs, Anna E. Fox, Nicholas R. Jungwirth, Raegan L. Johnson‐Wilke, Paul D. Dresselhaus, Samuel P. Benz

Published Jul 16, 2025• 0 citations

Abstract

This article presents a new multiimpedance-state line (MISL) in situ scattering parameter (S-parameter) calibration technique using on-chip superconducting transmission lines at 4 K that enables cryogenic calibration in a fixed signal path without the need for cryogenic switches or a cryogenic probe station. The method uses coplanar waveguide (CPW) models based on various impedance states of niobium (Nb), which has zero dc resistance below 9 K and a monotonically increasing resistance from 10 K to room temperature. The different impedance states are accessed by heating the 4 K stage of a cryostat and injecting up to 245 mA of current into the line. Using these states, we solve for the unknowns in an eight-term error model through a least-squares analysis. We first validate the MISL calibration technique by comparing it with short-open-load-reciprocal (SOLR) calibrated measurements in a cryogenic probe station, finding transmission agreement within 0.2 dB and uncertainty overlap for nearly all frequencies up to 26.5 GHz. We then apply the method to calibrate Nb CPWs with and without embedded Josephson junctions (JJs), using a fixed wire bonded connection, and without the use of cryogenic switches or movable probes. Strong agreement with the CPW models is demonstrated, with uncertainty overlap and differences below 0.1 dB up to 4.6 GHz without JJs and up to 2.4 GHz with JJs; resonances cause interruptions beyond these frequencies.

DOI: 10.1109/tmtt.2025.3585803

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We present the RF characterization of 2-µm-wide superconducting coplanar waveguides 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 4.7K–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/

\square

, a critical current density of 50 MA/cm

^{2}

, and a critical temperature of 8.9 K for Nb as well as 4.27 pH/

\square

, 3 MA/cm

^{2}

, 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.

Accurate in situ calibration of microwave attenuation and amplification-chain noise is essential for superconducting quantum circuits. We demonstrate a compact, self-calibrating cryogenic noise source based on an on-chip chromium attenuator that can be resistively heated with nanowatt-level power and directly integrated into a coaxial microwave line at the mixing-chamber stage. By comparing Johnson-Nyquist noise generated by Joule and microwave heating, measured through the amplification chain, the attenuation of the input line, and hence the gain of the chain, is determined without requiring knowledge of the attenuator temperature. The device exhibits millisecond-scale response times and negligible heating of the cryostat base plate. Using this approach, we determine the gain and added noise of a cryogenic amplification chain over the 4-8 GHz band. Our results provide a simple and accurate method to characterize near-quantum-limited parametric amplifiers used in superconducting-qubit readout.

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A method is proposed and experimentally explored for in-situ calibration of complex transmission data for superconducting microwave resonators. This cryogenic calibration method accounts for the instrumental transmission response between the vector network analyzer reference plane and the device calibration plane. Once calibrated, the observed resonator response is analyzed in detail by two approaches. The first, a phenomenological model based on physically realizable rational functions, enables the extraction of multiple resonance frequencies and widths for coupled resonators without explicit specification of the circuit network. In the second, an ABCD-matrix representation for the distributed transmission line circuit is used to model the observed response from the characteristic impedance and propagation constant. When used in conjunction with electromagnetic simulations, the kinetic inductance fraction can be determined with this method with an accuracy of 2%. Datasets for superconducting microstrip and coplanar-waveguide resonator devices were investigated and a recovery within 1% of the observed complex transmission amplitude was achieved with both analysis approaches. The experimental configuration used in microwave characterization of the devices and self-consistent constraints for the electromagnetic constitutive relations for parameter extraction are also presented.