⚛️ quantum physics

Superconducting Spiral Inductors for RF Reflectometry: Operation at Elevated Temperatures and Magnetic Fields

This paper presents a systematic study of NbTiN spiral inductors under elevated temperatures and magnetic fields, establishing design metrics and a benchmarking framework to optimize their performance for scalable RF reflectometry in future quantum computing architectures.

Original authors: Euan Parry, Murat Cubukcu, Patrick Reuvekamp, Manoj Stanley, Jonathan D. Fletcher, Alessandro Rossi

Published 2026-07-02

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CC BY 4.0

Original authors: Euan Parry, Murat Cubukcu, Patrick Reuvekamp, Manoj Stanley, Jonathan D. Fletcher, Alessandro Rossi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a very faint whisper (a quantum computer's signal) in a noisy room. To hear it clearly, you need a special listening device called a resonator. In the world of quantum computing, this device often uses a coil of wire, known as an inductor, to tune into the right frequency.

For a long time, scientists used standard, off-the-shelf coils (like the ones you might find in a radio). But these are bulky and have "parasitic" side effects that make the whisper harder to hear. The solution? Superconducting spiral inductors. Think of these as tiny, flat, spiral-shaped tracks made of a special metal (NbTiN) that carries electricity with zero resistance when it's cold. They are like a high-speed, frictionless racetrack for electricity, allowing for a much clearer signal.

However, there's a catch. Future quantum computers won't just sit in the deep freeze of absolute zero; they might need to operate in slightly warmer environments (several degrees above absolute zero) and under strong magnetic fields to work efficiently. The big question this paper asks is: Do these delicate superconducting spirals survive and keep working well under these tougher conditions?

Here is what the researchers found, explained through simple analogies:

1. The "Two-Part" Mystery

When you heat up a superconductor or put it in a magnetic field, the signal from the device changes. It's like a guitar string going out of tune. The researchers wanted to know: Is the string changing because the wood (the shape) is warping, or because the metal (the material) is changing?

To solve this, they used a clever "two-pronged" approach:

Method A: They listened to the device's natural "hum" (resonance) to see how the pitch changed.
Method B: They directly measured the electrical "heaviness" (inductance) of the coil separately.

The Discovery: They found that the "tuning" changes were almost entirely due to the material changing, not the shape. Specifically, as the temperature rose, the superconducting metal started acting a bit more like a normal metal, making the coil "heavier" (increasing its kinetic inductance). This is like a runner putting on a heavy backpack; they still run the same track, but they move slower. The researchers confirmed that the shape of the spiral didn't change; it was just the internal physics of the metal shifting.

2. The "Vortex" Problem (Magnetic Fields)

When they applied a magnetic field, the device started to lose its "purity" (its quality factor dropped). Imagine a perfectly smooth ice rink suddenly getting covered in tiny, swirling whirlpools.

The researchers found that even a tiny tilt in the magnetic field (like a slight wobble in the wind) caused these whirlpools, called vortices, to form at the edges of the spiral tracks. These vortices act like friction, slowing down the signal and creating noise.

The Fix: They discovered that wider tracks are more vulnerable to these whirlpools. It's like trying to keep a wide river calm in a storm; it's harder than keeping a narrow stream calm. Narrower tracks held up better against the magnetic field.

3. The "Goldilocks" Design Trade-off

The paper concludes with a set of rules for building these devices, which is a bit like designing a car:

Narrower tracks make the device smaller and more resistant to magnetic fields (good for the "storm").
However, making the tracks too narrow or using materials that are "too super" makes the device very sensitive to temperature changes. It's like a sports car that handles corners perfectly but stalls if the engine gets too hot.

The Bottom Line:
The researchers proved that these superconducting spirals are robust enough to work in the "warm" and "magnetic" conditions required for future quantum computers. They provided a blueprint for engineers to balance three competing needs:

Size: How small can we make it?
Stability: How much does the temperature mess it up?
Strength: How much magnetic field can it handle before it breaks?

By understanding exactly how the metal behaves under stress, they showed that we can design these tiny spirals to be the perfect, compact, and stable "listening ears" for the next generation of quantum technology.

Technical Summary: Superconducting Spiral Inductors for RF Reflectometry at Elevated Temperatures and Magnetic Fields

Problem Statement
Radio-frequency (RF) reflectometry is a standard readout technique for semiconductor spin qubits, typically employing a resonant tank circuit. While commercial surface-mount inductors are common, they suffer from large footprints, significant parasitic capacitance, and internal losses that limit readout sensitivity. Superconducting spiral inductors offer a compact, lithographically defined alternative with suppressed dissipative losses and tunable kinetic inductance. However, future scalable quantum architectures are expected to operate at elevated temperatures (above 1 K) and in magnetic fields of several hundred millitesla to leverage increased cooling power and integrate cryogenic control electronics. These conditions introduce new challenges: elevated temperatures increase quasiparticle populations and modify kinetic inductance, while magnetic fields can induce vortices and associated microwave dissipation. There is a lack of systematic data characterizing how these factors degrade the performance of superconducting spiral inductors, specifically regarding the separation of inductive and capacitive contributions to resonance shifts and quality factor ( $Q_i$ ) degradation.

Methodology
The authors conducted a systematic study of NbTiN spiral inductors (devices D1–D6) fabricated on high-resistivity silicon substrates. The devices featured varying track widths, turn numbers, and diameters, targeting inductances in the tens to hundreds of nH range. The study employed a dual-measurement approach to decouple inductive and capacitive effects:

Set-up 1 (Weakly Coupled Resonator): The spiral chip was positioned above a coplanar waveguide (CPW) to form a notch-type resonator. This allowed for the extraction of the intrinsic resonant frequency ( $f_0$ ) and internal quality factor ( $Q_i$ ) with minimal external loading.
Set-up 2 (Two-Port Inductance Extraction): The spiral was wire-bonded between two RF ports to measure S-parameters, enabling the independent extraction of inductance ( $L$ ) and series resistance over a low-frequency band where the structure behaves as a lumped element.

Measurements were performed in a Quantum Design PPMS over a temperature range of 2 K to 8 K and magnetic fields up to 750 mT (oriented nominally parallel to the film plane). An Unknown-Thru-Open-Short-Match (UOSM) calibration was used to de-embed spurious contributions from the measurement apparatus.

Key Contributions and Results

Separation of Inductive and Capacitive Effects: By combining the independent inductance extraction (Set-up 2) with resonator frequency measurements (Set-up 1), the authors demonstrated that temperature- and field-induced resonance shifts are predominantly inductive in origin. Using a constant self-capacitance ( $C_{self}$ ) derived from the data, the calculated frequency shifts based on measured inductance changes ( $L(T)$ and $L(B)$ ) matched the experimentally observed shifts with high fidelity. This confirms that $C_{self}$ remains largely unaffected by temperature and magnetic field variations in this regime.
Kinetic Inductance Characterization: The temperature dependence of inductance was fitted to a model incorporating the temperature-dependent sheet kinetic inductance ( $L_{k,\square}$ ). The extraction yielded a zero-temperature sheet kinetic inductance of $1.74$ pH/sq, a superconducting gap $\Delta_0 = 2.2$ meV, and a critical temperature $T_c = 13.4$ K. The results confirmed that the increase in inductance with temperature is driven by the reduction in superfluid density and the consequent increase in kinetic inductance.
Microwave Loss Mechanisms:
- Magnetic Field: The internal quality factor ( $Q_i$ ) remained stable at low fields but degraded sharply above a device-dependent onset field ( $B_Q$ ). This degradation is attributed to vortex entry driven by small residual perpendicular field components caused by experimental misalignment. The onset field $B_Q$ was found to scale inversely with track width ( $w$ ), suggesting that wider tracks are more susceptible to vortex nucleation.
- Temperature: In zero field, $Q_i$ decreased with increasing temperature, consistent with a phenomenological quasiparticle-loss model. The data suggests the low-temperature loss limit for these devices is well below 2 K.
Design Metrics: The study established practical design metrics linking geometry to performance:
- Kinetic Inductance Fraction ( $\alpha$ ): Defined as $L_k/(L_g + L_k)$ , $\alpha$ dictates the sensitivity of the resonance frequency to temperature. While higher $\alpha$ (achieved via narrower tracks or higher disorder) allows for greater miniaturization, it increases frequency sensitivity. However, all measured devices exhibited frequency shifts below 1% between 2 K and 8 K.
- Field Robustness: The magnetic field tolerance ( $B_Q$ ) is a function of track width, with narrower tracks offering better resilience against vortex entry under misaligned fields.

Significance and Claims
The paper claims to provide a general framework for benchmarking superconducting inductors by separating inductive and capacitive contributions to device behavior. The authors assert that their results demonstrate NbTiN spiral inductors can maintain high quality factors ( $>10^3$ ) and stable operation in the elevated temperature and magnetic field regimes targeted for scalable spin-qubit architectures.

The work positions NbTiN spirals as occupying an "intermediate design space" between standard commercial inductors and more aggressive high-kinetic-inductance approaches (e.g., granular films or superinductors). They offer a balance of compactness, low loss, and modest kinetic inductance participation, resulting in small frequency shifts over the measured temperature range. The authors conclude that these inductors are attractive for compact RF-reflectometry circuits, provided that the trade-offs between footprint, frequency stability, and magnetic-field resilience are carefully managed through geometric design (specifically track width and kinetic inductance fraction). The study does not claim to solve all challenges for high-field operation but provides the necessary metrics to guide the selection of inductor geometries for specific quantum computing environments.

1. The "Two-Part" Mystery

2. The "Vortex" Problem (Magnetic Fields)

3. The "Goldilocks" Design Trade-off

Technical Summary: Superconducting Spiral Inductors for RF Reflectometry at Elevated Temperatures and Magnetic Fields

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