Liquid metal printing for superconducting circuits

TL;DR

The paper addresses the challenge of fabricating low-loss superconducting circuits in a scalable, additive manner. It introduces capillary-based liquid-metal printing of the eutectic GaInSn alloy to pattern lumped-element resonators on sapphire substrates. Key findings show single-photon quality factors of $Q_i \approx 6\times10^5$ at resonant frequencies around $f_0 \approx 5.8$ GHz, with potential to approach $Q_i$ near $10^6$ under favorable conditions, indicating competitive coherence with lithographic approaches. The study also reveals reliability challenges: repeated thermal cycling induces destructive phase transitions in the liquid metal (e.g., tin pest), linked to microstructural phase separation observed via EDS and optical cryo-microscopy, highlighting the need for materials engineering to enable robust operation. Overall, the method offers a path to additive, locally addressable superconducting circuitry for scalable quantum hardware, contingent on mitigating phase-transition–related degradation through alloy design or alternative inks.

Abstract

Superconducting circuits are a promising platform for implementing fault-tolerant quantum computers, quantum limited amplifiers, ultra-low power superconducting electronics, and sensors with ultimate sensitivity. Typically, circuit fabrication is realized by standard nanolithography, generally associated with a high level of control over defects and contaminants. Additive approaches have not been used so far since they are expected to be inferior in terms of superconducting properties or quantum coherence. This work shows that liquid-metal based micro-pipette printing is suited for fabricating superconducting lumped-element resonators with high internal quality factors. The applicability of our technique for low-loss superconducting device fabrication and the possibility to locally add metal structures, without affecting any preexisting circuit elements, is a further advantage. Our results open up new avenues in the hardware implementation of scaled-up superconducting quantum computers.

Figures (13)

• Figure 1: Liquid metal capillary printing of superconducting lumped-element resonators.(a) Illustration of the capillary printing setup used for fabricating superconducting resonators. The inset on the left shows a photograph of lines printed using this setup with the liquid metal alloy EGaInSn. The right-hand inset depicts the capillary tip during printing as seen through the optical microscope. (b) Optical micrograph of an EGaInSn lumped-element resonator printed with the setup shown in (a). The inset shows the lower left corner of the resonator at higher magnification, highlighting the achieved linewidth and pitch. The gray arrows specify the order and directions in which the resonator lines were printed. (c) Plots of the electric field magnitude $E_z$ (blue and red color scale) and magnitude of the current density $|j|$ (green color scale) derived from electro-magnetic finite-element eigen-mode simulation for the fundamental mode at $f_0 \approx 5.5GHz$. The plots correspond to the phases at which field and current are at a maximum, respectively. They show that the circuit can be interpreted as a lumped-element resonator with the equivalent circuit shown in the inset and $f_0 = 1/(2\pi\sqrt{L(C_1+C_2)})$.
• Figure 2: Cryogenic measurement of the single-photon resonator quality factor $Q_\mathrm{i}$.(a) Schematic of the dilution cryostat measurement setup used to characterize microwave loss of the printed resonators at millikelvin temperature. The resonators were mounted in an aluminum waveguide, as depicted in the photograph with the waveguide lid opened to reveal the resonator sample. It was attached to the mixing chamber stage of the cryostat. The plots in (b) and (c) show the reflected amplitude $|S_{11}|$ and phase $\mathrm{arg}(S_{11})$ measured with the vector network analyzer (VNA) for Resonator 1 near its resonance frequency $f_0 = 5.791625GHz$ and the lowest microwave power, resulting in resonator photon occupation $\bar{n}\approx1$ and thus most relevant for quantum applications. In (d) the reflected signal is shown to follow a circular path in the complex plane. The dashed lines show the harmonic model fit for $S_{11}$ used to extract the internal quality factor $Q_\mathrm{i}$. The latter depends on the resonance width $\Delta f$ and circle radius $r$ as specified in the formula. (e) Dependence of $Q_\mathrm{i}$ on the exciting microwave power, shown as the derived average photon number $\bar{n}$ stored in the resonator, for three printed resonators (Resonator 1 - 3). The shaded region indicates uncertainty from Fano interferenceRieger2023Fano.
• Figure 3: Superconducting critical temperature and effects of cool-down and warm-up on Resonator 1.(a) Frequency shift $\Delta f(T) = f(T)-f_0$ and quality factor $Q_\mathrm{i}(T)$ for Resonator 1 as a function of temperature. The solid lines are a fit to a model for surface impedance due to thermal quasi-particles yielding EGaInSn's superconducting critical temperature $T_\mathrm{c}$. (b) Energy-dispersive X-ray elemental map of the cross-section of a printed EGaInSn line at around 88K reveals a segregation of gallium (red) from indium (green) and tin (blue). (c) Optical micrograph of Resonator 1 after the first cryogenic measurement. Markedly, the uppermost printed trace has changed in length likely due to dewetting. Where the liquid metal has retracted, some residues remain which have possibly alloyed with the substrate. The inset shows that the printed traces have increased in surface roughness with intermittent constrictions.
• Figure 4: Destructive effect of thermal cycling on printed traces.(a) Optical micrograph of Resonator 2 after first cool-down/warm-up. The insets show affected parts of the resonator at higher magnification. Some lines were fully detached from the substrate while at other locations more local defects occurred. (b) Optical cryo-microscopy experiment with a printed resonator. This was conducted in a separate helium flow cryostat probe station with an optical access window, in order to observe the printed structure during cool-down and warm-up. The left-most image shows the full resonator structure at room-temperature as mounted in the cryostat. The colored segments indicate the temperature ranges in which the recorded micrographs did not change. Changes associated with the solid/liquid phase transformation occurred at 218K and 283K and a destructive change was observed during the cooling at about 23K.
• Figure 5: Different printing approach that was used to fabricate Resonators 2 and 3. The Resonators 2 and 3 were fabricated using the printing approach in (a), where the substrate and capillary tip remain in contact during the entire printing process. The designed Resonator structure cannot be fully printed in a single pass without lifting the capillary. Therefore, in order to ensure galvanic contact we decided to retrace the lines, eventually returning to the initial point. The selected printing path is as shown with common start and end point. In this approach, the lines were printed in 100µ m-steps. At the corners of the resonator, the tip was lifted by approximately 5µ m from the substrate and then lowered again by the same amount to avoid any drifting of the capillary tip. This is done similarly when retracing the previously printed lines. The capillary holder allows to adapt the contact angle in x-direction, which is usually set to roughly 5060°, permitting good adhesion of the liquid metal to the substrate. Furthermore, printing in y-direction, which was done in this approach, is more difficult. It negatively impacts line reproducibility and makes discontinued lines more likely, since the contact angle in y-direction is fixed to 90, and cannot be adjusted. When retracing, the tip was lifted by around 10µ m (not disconnecting capillary flow) to prevent displacing the already printed line. With this mostly manual approach the yield of the printed resonators was low. The optical micrographs in (b) and (c) show Resonator 2 and 3 respectively.