# Universal scaling of microwave dissipation in superconducting circuits

Thibault Charpentier, Anton Khvalyuk, Lev Ioffe, Mikhail Feigel’Man, Nicolas Roch, Benjamin Sacépé

► To cite this version:

Thibault Charpentier, Anton Khvalyuk, Lev Ioffe, Mikhail Feigel’Man, Nicolas Roch, et al. Universal scaling of microwave dissipation in superconducting circuits. 2025. hal-05219918

HAL Id: hal-05219918

https://hal.science/hal-05219918v1

Preprint submitted on 22 Aug 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
Universal scaling of microwave dissipation in superconducting circuits

Thibault Charpentier,¹, * Anton Khvalyuk,², * Lev Ioffe,³

Mikhail Feigel’man,⁴,⁵ Nicolas Roch,¹ and Benjamin Sacépé¹, †

¹Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France

²Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France

³Google Quantum AI, Mountain View, CA

⁴Jožef Stefan Institute, Ljubljana 1000, Slovenia

⁵CENN Nanocenter, Ljubljana 1000, Slovenia

(Dated: July 15, 2025)

Improving the coherence of superconducting qubits is essential for advancing quantum technologies. While superconductors are theoretically perfect conductors, they consistently exhibit residual energy dissipation when driven by microwave currents, limiting coherence times. Here, we report a universal scaling between microwave dissipation and the superfluid density, a bulk property of superconductors related to charge carrier density and disorder. Our analysis spans a wide range of superconducting materials and device geometries, from highly disordered amorphous films to ultra-clean systems with record-high quality factors, including resonators, 3D cavities, and transmon qubits. This scaling reveals an intrinsic bulk dissipation channel, independent of surface dielectric losses, that originates from a universal density of nonequilibrium quasiparticles trapped within disorder-induced spatial variations of the superconducting gap. Our findings define a fundamental limit to coherence set by intrinsic material properties and provide a predictive framework for selecting materials and the design of next-generation superconducting quantum circuits.

Introduction

The development of scalable and fault-tolerant superconducting quantum processors hinges on the ability to maximize the coherence time of qubits, which directly impacts the fidelity and efficiency of quantum computations [1, 2]. Despite continuous advancements in the last two decades, the coherence time typically falls in the range of tens of microseconds to a few milliseconds [3–7]. This limitation stems from energy dissipation mechanisms that are not yet fully understood but are believed to arise from both intrinsic and extrinsic sources in materials, circuit designs and radiations [8–10].

The fundamental challenge lies in the fact that even pristine superconductors are found to lose energy when carrying an oscillating microwave current near absolute zero temperatures. These losses are characterized by the internal quality factor, $Q_i$, the ratio between the energy stored and the energy loss per cycle of oscillation, which inversely relates to the dissipation rate. In qubits, this dissipation translates into a finite coherence time $T_1$, linked by the relation $Q_1 = \omega T_1$ where $\omega/2\pi$ is the qubit frequency.

A considerable body of work has focused on identifying and quantifying the mechanisms responsible for this residual dissipation across various superconducting materials. These mechanisms include dielectric losses [11], non-equilibrium quasiparticles [12–15], magnetic vortices [16], radiative losses [8–10], and, more recently,

mechanical vibrations [17] or substrate piezoelectricity [18, 19]. Among these, dielectric losses –particularly those arising from defects that form a bath of two-level systems (TLS) at material surfaces and interfaces– have garnered significant attention [11]. Their ubiquitous presence and universal character in most dielectrics and oxidized surfaces [20, 21], and their remarkable ability to qualitatively explain the diverse behaviors observed with increasing excitation power or temperature [11], makes them central to most analyses as the dominant source of dissipation.

However, a class of materials, such as granular aluminum (grAl) [22–24], titanium nitride (TiN) [25, 26], amorphous indium oxide (a:InO) [27], or tungstene silicide (WSi) [28] known as strongly disordered superconductors [29], defies the dielectric loss paradigm. These materials lie near a disorder-tuned superconductor-insulator transition [29] and exhibit quality factors that decrease with disorder [24, 27] and reach values as low as ~ $10^3$ for the most disordered samples [27], only an order of magnitude higher than those reported for metallic resonators [30]. Their dissipation is found to be independent of the surrounding dielectric environment, suggesting it arises from intrinsic bulk mechanisms, sometimes referred to as inductive loss. This behavior contrasts sharply with clean elementary superconductors such as Al, Nb or Ta, which achieve $Q_i$ values up to $10^6 – 10^7$ in thin-film resonators [4, 6, 7, 31–33], and as high as $10^{10}$ in Nb cavities at low photon numbers and low temperatures [34].

Paradoxically, despite the bulk nature of dissipation in strongly disordered superconductors, their power and temperature dependencies [22, 26, 27] closely resemble

<footnote>* These two authors contributed equally</footnote>

<footnote>† benjamin.sacepe@neel.cnrs.fr</footnote>
those typically attributed to TLSs in clean materials.

This unusual bulk dissipation raises the question of whether static disorder plays a role in the intrinsic microwave dissipation in superconductors, in apparent contradiction with the standard electrodynamics theory of superconductivity [35]. The latter states that a superconductor, even in the presence of strong disorder, should not dissipate energy at frequencies below twice the superconducting gap.

In this work, we address this question through a systematic and comprehensive analysis of internal quality factors reported for conventional superconductors – including planar resonators, 3D cavities, and transmon qubits. By correlating dissipation with the inductive response, which reflects the degree of material disorder, we uncover a new fundamental limit to microwave coherence, set by bulk conductive losses driven by nonequilibrium quasiparticles.

## Superfluid density-dependent limit for microwave dissipation

We begin by introducing the inductive response, a fundamental quantity that reflects how disorder influences the electrodynamics of superconductors. This response is captured by the imaginary component, $\sigma_2(\omega)$, of the complex conductivity $\sigma(\omega) = \sigma_1(\omega) - i\sigma_2(\omega)$, which directly relates to the superfluid density, $n_s$, and to the magnetic penetration depth, $\lambda$, via

$$ \sigma_2(\omega) = \frac{n_s e^2}{m^* \omega} = \frac{1}{\mu_0 \omega \lambda^2}, \quad (1) $$

where $e$ and $m^*$ are the electron charge and effective mass, respectively, and $\mu_0$ is the vacuum permeability. Physically, the oscillating supercurrent that flows over the penetration depth exhibits a kinetic inductance $L_K = \mu_0\lambda$, which describes the kinetic energy stored by the motion of the Cooper pair condensate. For thin films of thickness $d < \lambda$, the magnetic field penetrates entirely and $L_K = \mu_0\lambda^2/d$.

In dirty superconductors, where the mean free path $l$ is smaller than the coherence length in the clean limit $\xi_0$, disorder reduces the superfluid density as $n_s \propto \pi n \Delta \tau / \hbar$, where $n$ is the electron density, $\Delta$ is the superconducting gap, and $\tau$ is the elastic scattering time [36]. Consequently, according to Eq. (1), $\sigma_2(\omega)$ decreases with increasing disorder, while the magnetic penetration depth increases [37]. This leads to an increased kinetic inductance and a weakened superfluid stiffness, which reflects a reduction in the overall superconducting phase rigidity [27].

Figure 1 presents the central result of this work: the evolution of quality factors $Q_i$ as a function of $\sigma_2$ for most of the studied materials available in the literature (see Table I). The dataset includes both thin-film planar resonators (filled circles), 3D cavities (empty triangles), and

transmon qubits (filled stars). To ensure consistent comparisons, $Q_i$ values for resonators are restricted to low photon numbers and the lowest experimentally accessible temperatures (typically below 0.1 K). This minimizes material-dependent photon-number or temperature dependencies, which, while sometimes non-monotonic, typically vary $Q_i$ by no more than an order of magnitude. Given the multi-decade variation in both $Q_i$ and $\sigma_2$ on the double-logarithmic scale and the scatter in the data, accounting for these dependencies would only result in a slight upward shift in $Q_i$ values, without altering our findings.

Two main striking observations emerge from this figure. First, the dataset is conspicuously divided by a diagonal line in the $\sigma_2 - Q_i$ plane, above which no values of the quality factor have ever been reported. All collected values of $Q_i$ appear to be bounded by the empirical relation $Q_i^{\max} \approx \kappa \sigma_2$ with a proportionality factor $\kappa \sim (0.1 - 1) \, \Omega.m$ (see diagonal red strips), which establishes an upper limit of $Q_i$ for a given material's $\sigma_2$. This boundary extends nearly linearly across the entire range of $\sigma_2$, that is, over seven order of magnitude, encompassing strongly disordered superconductors such as grAl, a:InO, and TiN/AlN, moderately disordered superconductors like Hf and TiN, and even includes 3D cavities of Al and Nb, which achieve record-high $Q_i$ and $\sigma_2$ values. For two-dimensional films, the boundary can be expressed as

$$ Q_{\text{film}}^{\max} = \frac{1}{\eta} \frac{h}{e^2} \sigma_2^{\square} = \frac{1}{\eta} \frac{h}{e^2} \frac{1}{L_K \omega}, \quad (2) $$

where $\sigma_2^{\square}$ is the sheet conductance and $\eta \sim 10^{-2}-10^{-3}$ a dimensionless constant. The microscopic origin of $\eta$ will be discussed later in the article.

Since most superconducting resonators and cavities operate within a narrow bandwidth of 2–12 GHz (with an average frequency of the entire dataset $\omega/2\pi = 5.5$ GHz, see Supplementary Materials Figure 1), $\sigma_2$ is primarily dependent on the material's superfluid density via the relation $\sigma_2 \propto n_s/\omega$.

Importantly, this dissipation limit is set by the best quality factors achieved for a given material or level of disorder, corresponding to a specific $\sigma_2$ value. However, a significant portion of the data lies below this boundary. These lower quality factors result from external dissipation channels that dominate and prevent the limit from being reached. These channels include standard dielectric losses, characterized by their loss tangent $\tan \delta_m$ and associated electric energy participation ratio $p_m$ (where $m$ labels the respective interface), as well as circumstantial factors such as material or sample processing imperfections. To quantify how these dissipation channels limit $Q_i$, we express the quality factor as: $\frac{1}{Q_i} = \sum_m p_m \tan \delta_m + \frac{1}{Q_{\text{MB}}} + \frac{1}{Q_{\text{qp}}} + \frac{1}{Q_{\text{others}}}$, where $Q_{\text{MB}}$ represents the Mattis-Bardeen contribution of thermal quasiparticles [35], which is typically negligible below 0.1 K, $Q_{\text{qp}}$ accounts for non-equilibrium quasiparticles, and
<figure><img src="image_2.png" /><figcaption>FIG. 1. Microwave dissipation from disordered to clean superconductors. Internal quality factor $Q_i$ of superconducting resonators and transmon qubits measured at low temperature ($T \lesssim 0.1$ K) and low power, plotted as a function of the imaginary part of the complex conductivity, $\sigma_2 = 1/\mu_0\omega\lambda^2$, for various superconducting materials. The values of the magnetic penetration depth $\lambda$, and hence $\sigma_2$, span several orders of magnitude across materials, reflecting differences in disorder levels and charge carrier densities in cleaner superconductors. Dots represent devices in 2D geometries (coplanar waveguides or microstrip resonators), stars denote 2D transmon qubits, crosses correspond to suspended striplines, and empty triangles represent 3D resonant cavities. The two diagonal red stripes highlight the upper bound on microwave quality factors due to localized quasiparticles given by Eq. (3). For $\sigma_2 \lesssim 10^8$ S.m, the resulting dissipation follows the scaling $Q_i^{\text{max}} = \kappa \sigma_2$ with a proportionality factor $\kappa \approx (0.1 - 1) \, \text{\Omega.m.}$ For $\sigma_2 \gtrsim 10^8$ S.m, the diagonal strip reflects the scaling $Q_i^{\text{max}} \propto \sigma_2^{3/2}$, specific for small kinetic inductance fraction. The dashed gray line is a visual guide with slope of 3/2, fitting the best-performing cavities. A horizontal line at $Q_i \approx 10^7$ marks an apparent additional limit between planar devices (blue area) and 3D cavities (orange area), most likely related to dielectric substrate losses. Data and references are provided in Supplementary Materials Table I.</figcaption></figure>
limiting $Q_i$, a common strategy to mitigate their impact consists of reducing $p_m$ by diluting electric field with large circuits. A body of work on clean superconductors has shown that reducing $p_m$ indeed enhances $Q_i$ [38–42], up to a critical dimension beyond which $Q_i$ ceases to improve and saturates [33, 43]. This trend is readily seen in Figure 1 where vertically aligned clusters of data points for a given material ($\sigma_2$) illustrate this continuous increase in $Q_i$ with increasing sample dimensions, corresponding to a reduction in $p_m$ (see for instance TiN, Hf, Ta).

In stark contract, experiments on strongly disordered films such as grAl [22, 24], a:InO [27, 44], TiN [26, 45], and WSi [28] show no correlation between $Q_i$ and participation ratios, thereby ruling out dielectric loss as the dominant dissipation mechanism in the presence of disorder.

## Dissipation from nonequilibrium quasiparticles

The upper limit of the quality factor, observed at low photon numbers and low temperatures, scales with the material’s complex conductivity –or equivalently, its superfluid density. This empirical law, which holds across both 2D circuits and 3D cavities and spans materials from strongly disordered to clean superconductors, effectively rules out dielectric loss as the dominant dissipation mechanism and provides compelling evidence for a bulk origin of dissipation –namely, conductive loss.

As a phenomenon common to all superconducting circuits, dissipation due to non-equilibrium quasiparticles emerges as a likely candidate [12–15, 46]. Various sources of such quasiparticles have been identified, including high-energy stray photons propagating through wiring [47–53], mechanical vibrations [54] as well as cosmic radiation and radioactivity, which induce high-energy phonon down-conversion via pair-breaking [55–59]. Inadequate sample thermalization, a more trivial issue, could also be a contributing factor. Additionally, material imperfections and surface conditions can generate localized magnetic moments and associated sub-gap states [60, 61].

Since the latter are material-specific, they can be excluded as a common source. Cosmic radiation and environmental radioactivity, while universal, produces sparse quasiparticle bursts due to its slow event rate. On the other hand, most, if not all, datasets reported in Fig. 1 were likely obtained under similar experimental conditions, using comparable microwave components (wires, attenuators, filters, and amplifiers), many of which are now standard commercial products. This suggests that the power spectrum of unfiltered stray photons propagating through wiring and packaging is broadly similar across state-of-the-art experiments at the diagonal dissipation limit. Consequently, the quasiparticle generation rate due to pair-breaking should also be comparable.

The central question is thus the fate of these non-equilibrium quasiparticles and their contribution to dissipation. Once generated, quasiparticles rapidly relax

to the superconducting gap edge, where they become trapped and recombine inefficiently. This trapping is due to sub-gap states near the gap edge, which inevitably arise in superconductors due to disorder [62–64]. In the strong-disorder limit, these sub-gap states result from substantial spatial fluctuations of the superconducting gap [29], as evidenced by scanning tunneling spectroscopy on thin films of various disordered materials, including TiN [65], NbN [66], a:InO [67], and MoGe [68], forming shallow trapping potential wells where the gap is locally suppressed [69, 70].

A recent theory predicts that the density of trapped quasiparticles scales inversely with the superconducting coherence volume, $n_{qp} \propto 1/\xi^3$ or $n_{qp} \propto 1/(\xi^2 d)$ when $\xi > d$, and only logarithmically with both the quasiparticle generation rate and the strength of electron-phonon relaxation [69]. In realistic materials, one can reasonably assume that the inter-quasiparticle distance –beyond which localized quasiparticles do not overlap and hence fail to recombine– cannot be much larger than $10\xi$ (see Supp. Mat.). This implies a universal dimensionless quasiparticle density $n_{qp}\xi^2 a = \eta$, with $\eta \gtrsim 0.01–0.001$ and $a = \min(\xi, d)$, which is in good agreement with experimental values of residual quasiparticles [14, 71–73].

The microscopic contribution of trapped quasiparticles to dissipation –specifically, to the real part of the complex conductivity $\sigma_1(\omega)$– remains poorly understood. Yet, in the case of disordered superconducting films, a simple estimate can be derived under the assumptions that localized quasiparticles are ionized into the continuum by microwave absorption. By extending the Mattis-Bardeen formula to nonequilibrium localized quasiparticles [35, 61], one obtains a universal and constant real part of the conductivity, $\sigma_1 \sim \sigma_n n_{qp}/(\nu_0\Delta) \sim \eta e^2/(a\hbar)$, where the conductivity $\sigma_n$ is approximated by the Drude value, $\sigma_n = e^2\nu_0 v_F l/3$, and the coherence length is given by $\xi^2 = v_F l\hbar/(6\Delta)$. Here, $\nu_0$ is the density of states, and $v_F$ the Fermi velocity. This leads to:

$$Q_{qp} = \frac{1}{\eta\alpha} \frac{h}{e^2} a\sigma_2, \quad (3)$$

where $\alpha = L_K/(L_K + L_g)$ is the kinetic inductance fraction with $L_g$ the geometric inductance. Remarkably, this result recovers the empirical trend captured by Eq. (2). Since the coherence length $\xi$ typically varies within a factor of three across disordered materials –usually 5 nm $\lesssim \xi \lesssim 15$ nm– and $\alpha \lesssim 1$ for devices with large kinetic inductance, this implies $Q_{qp} \sim 0.2 \, \Omega \cdot m \times \sigma_2$ (with $a = \xi \sim 10$ nm and $\eta \sim 10^{-3}$). This is in excellent agreement with the observed scaling in the low-$\sigma_2$ regime in Fig. 1.

In the Supp. Mat., we derive more accurate estimates for $\sigma_1$ under various dissipation mechanisms (see Supplementary Materials Fig. 2) by combining realistic distribution functions for localized quasiparticles with disorder-broadened density of states [62, 74]. This framework provides a comprehensive classification of quasiparticle-
induced losses and highlights open theoretical questions (see Suppl. Mat. Fig. 3). Notably, our more elaborated estimate of the prefactor agrees well with the diagonal trend in Fig. 1 (see Eq. (21) in Supp. Mat.).

Extending Eq. (3) to clean superconducting cavities with large $\sigma_2$ values leads to a distinct scaling, $Q_i \propto \sigma_2^{3/2}$, due to the small kinetic inductance fraction $\alpha \sim \omega\lambda/c \ll 1$, which is dominated by geometric inductance ($c$ is the speed of light). Remarkably, the best-performing cavities in Fig. 1 align with the corresponding slope 3/2, shown by grey dotted line, consistent with conductive loss. We emphasize, nonetheless, that applying the localized quasiparticle model to the clean limit predicts quality factors indicated by the upper red strip in Fig. 1, significantly exceeding the existing data –even though it correctly captures the dependence on $\sigma_2$ related to conductive loss. Therefore, attributing the observed $Q_i \propto \sigma_2^{3/2}$ scaling in the best-performing cavities to localized quasiparticles requires further investigation.

An open question concerns the power dependence of resonator quality factors, which is commonly attributed to the saturation of TLS baths, and appears somewhat at odds with the quasiparticle-based interpretation. It has been shown that trapped quasiparticles can, in some circumstances, be described as TLS [70], suggesting a similar increase in $Q$ with drive power. Moreover, recent experiments on grAl devices [22] showed that their quality factor increases with power in a manner typically attributed to dielectric TLSs, yet exhibits no dependence on the electric field participation ratio. These observations suggest that quasiparticles, rather than dielectric TLSs, may dominate both dissipation and its power dependence –a possible reinterpretation that has received little experimental attention to date. Ongoing theoretical work [48, 61, 69] further supports this perspective by addressing the kinetics of non-equilibrium quasiparticles and proposing specific dissipation mechanisms, such as microwave-assisted overheating or ionization.

Another ubiquitous source of dissipation arises from vortices, which may enter superconductors during cooldown if residual magnetic fields are present. Estimating the vortex-induced dissipation, assuming optimistically low residual magnetic fields, leads to a quality factor scaling with $\sigma_2$, but with a strongly non-universal prefactor that cannot account for the observed diagonal limit (see Supp. Mat.).

## Substrate loss in planar circuits

The second striking observation that emerges from Figure 1 is the presence of another limit specific to planar resonators: for clean superconductors with $\sigma_2 \gtrsim 10^8$ S.m$^{-1}$, most quality factors saturate at $Q_i \sim 10^7$, independent of their $\sigma_2$ value. This limit is highlighted by the horizontal boundary that separates the orange and blue area in Fig. 1. It corresponds to the saturation of quality factors upon minimizing electric-field energy in

dielectric interfaces.

The emergence of this limit suggests a common dissipation channel for planar geometry, ruling out material-dependent interface losses. Two dissipation mechanisms that are common to all microwave circuits are radiation loss and substrate loss.

As samples are most often packaged in tightly sealed (super)conductive boxes with dimensions smaller than the free-space wavelength, far-field radiation is likely suppressed, and therefore unlikely to be responsible for the limit.

On the other hand, substrate loss in commonly used silicon and sapphire substrates has recently been thoroughly studied. For high-resistivity silicon wafers, $\tan\delta$ has been reported to be as low as $2.5 \times 10^{-7}$ [41, 42], while a recent study measured an order of magnitude higher $2.2 \times 10^{-6}$ [75]. We emphasize that comparing silicon substrate dissipation mechanisms is challenging due to variation in wafer resistivity [76]. Moreover, the impact of intentionally introduced dopants used to homogenize wafer resistivity within the $10-20$ kΩ.cm range, or other impurities (e.g. oxygen or carbon) on microwave dissipation remains poorly understood. Thus, dissipation in silicon substrates, with these above assessments should limit the quality factor to approximately $Q_i \sim 5 \times 10^6$, assuming $p_{\text{Substrate}} \simeq 0.8$. This estimate is consistent with recent work that achieved $Q \sim 10^7$ on a silicon substrate [7], confirming that the exact value of loss tangent can slightly vary depending on the wafer source. For sapphire, two substrate grades obtained via different growth Supp. Mat. have been thoroughly studied under various processing and annealing conditions [4, 77]. These studies report $\tan\delta$ values ranging from 2 to $35 \times 10^{-8}$, corresponding to a substrate-limited quality factor between $3 \times 10^6$ and $6 \times 10^7$ [4].

Clearly, bulk dissipation in silicon and sapphire substrates matches the limit reached by planar superconducting microwave circuits. A recent study [4] further supports this finding, demonstrating that optimized resonator geometries, designed to minimize surface losses on the highest grade, annealed sapphire substrates, have exceeded this limit, achieving record $Q_i$ values up to $\sim 3 \times 10^7$ (see Ta resonators at $\sigma_2 = 10^9$ S.m$^{-1}$ in Fig. 1).

## Transmon qubits

Interestingly, transmon qubits, which have a radically different geometry from resonators, also fall below this horizontal limit. In Fig. 1, we include quality factor data for transmon qubits (see filled stars), inferred from their energy relaxation times ($T_1$), by linking them to the $\sigma_2$ of their capacitor electrodes. Although $\sigma_2$ is not strictly defined for transmons, we assume that the universal limit on $Q_i$ observed in resonators may also constrain transmon coherence via losses in the capacitor and wiring electrode materials –predominantly in the case of disordered, inductive superconductors, or marginally for
clean superconductors– alongside other loss mechanisms. This contribution is evident in transmons made from disordered materials, as exemplified by GrAl transmons for which $Q_i \sim 10^5$ [78, 79] (see filled green stars in Fig. 1).

For transmons made with clean superconductors (e.g. Al, Ta, Nb, TiN), most $Q_i$ values do not reach the diagonal limit and remain bounded at $\sim 10^7$, closely matching the upper limit observed in planar resonators (see Fig. 1). As thoroughly shown in Refs. [7, 45], transmon coherence time improves as $p_m$ of interfaces decreases, until it saturates at a value similar to that of resonators [80, 81]. Several explanations have been proposed, including intrinsic dissipation within the Josephson junction [45] or losses in its surroundings due to enhanced local electric fields [4]. However, the coincidence with the $Q_i \sim 10^7$ limit observed in resonators strongly suggests a shared dissipation mechanism, ruling out the Josephson junction and its dielectric environment as dominant loss sources, and instead points to bulk substrate loss as the underlying limitation. This conclusion, which emerges from Fig. 1, is in full agreement with a recent loss budget analysis performed on state-of-the-art transmons [4].

## Implications for Quantum Circuits

Our analysis, based on the most comprehensive dataset currently available in the literature, reveals an intrinsic upper limit to the coherence of quantum circuits for each superconducting material accessible with current technology. The dependence of quality factor on superfluid density allows us to identify niobium as the optimal material, exhibiting the highest superfluid density, as demonstrated by 3D cavities achieving record-long photon lifetimes on the order of seconds [34]. Assuming resonant circuits operate at this upper limit, for widely used aluminum-based technology, or even tantalum, maximum coherence time will lag by an order of magnitude compared to niobium. Moreover, as our analysis shows, this limit originates from residual quasiparticles, substantial improvements in filtering [46, 56, 82, 83], gap engineering [84–87], phonon trapping [55, 88–90] and packaging [59] will be essential to push this coherence limit further.

The above arguments assume effective mitigation of interface-related dissipation and do not account for substrate dissipation, which we identify as another limiting factor for 2D planar circuits. Addressing substrate dissipation will require progress on materials quality or the development of radically new architectures, such as

substrate-free suspended circuits or on-chip 3D designs.

At the individual qubit level, our findings indicate that coherence times on the diagonal limit could, in principle, reach $T_1 \sim 0.1-1$ s, which would represent a transformative milestone for quantum computing. However, scaling up to multi-qubit processors [91] inevitably introduces complex couplings and additional dissipative channels. Moreover, the associated advanced fabrication techniques impose further constraints on material and interface quality. Taken together, these challenges make reaching the ultimate coherence limit extremely demanding for multi-qubit architectures. Looking ahead, our results establish a fundamental benchmark that defines the ultimate coherence goal within the industry's roadmap for superconducting quantum computing, which is rapidly advancing and holds great promise [92].

## DATA AVAILABILITY

The code and dataset used to plot Fig. 1 are available at Zenodo 10.5281/zenodo.15737011.

## ACKNOWLEDGMENTS

We thank D. Basko, Q. Ficheux, R. Gao, L. Glazman, M. Houzet, J. Meyer, I. Pop, and M. Scheffler for valuable discussion. We thank R. Gao, N. de Leon, and I. Pop for sharing their data. B.S. thanks Y. Chen for many valuable discussions. B.S. and T.C. has received funding from the European Union's Horizon 2020 research and innovation program under the ERC grant SUPERGRAPH No. 866365. N.R. has received funding from the European Union's Horizon 2020 research and innovation program under the ERC grant SuperProtected No. 101001310. N.R. and B.S. acknowledge funding from the ANR agency under the 'France 2030' plan, with Reference No. ANR-22-PETQ-0003. A.V.K. is grateful for the support by Laboratoire d'excellence LANEF in Grenoble (ANR-10-LABX-51-01).

## COMPETING INTERESTS

The authors declare that they have no competing interests.

[1] M. H. Devoret and R. J. Schoelkopf, *Science* **339**, 1169 (2013).

[2] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson, and W. D. Oliver, *Applied Physics Re-

views* **6**, 021318 (2019).

[3] M. Kjaergaard, M. E. Schwartz, J. Braumüller, P. Krantz, J. I.-J. Wang, S. Gustavsson, and W. D. Oliver, *Annual Review of Condensed Matter Physics* **11**,
369 (2020).

[4] S. Ganjam, Y. Wang, Y. Lu, A. Banerjee, C. U. Lei, L. Krayzman, K. Kisslinger, C. Zhou, R. Li, Y. Jia, M. Liu, L. Frunzio, and R. J. Schoelkopf, Nature Communications 15, 3687 (2024).

[5] M. Tuokkola, Y. Sunada, H. Kivijärvi, J. Albanese, L. Grönberg, J.-P. Kaikkonen, V. Vesterinen, J. Gove- nius, and M. Möttönen, arXiv:2407.18778 (2024).

[6] A. Dane, K. Balakrishnan, B. Wacaser, L.-W. Hung, H. Mamin, D. Rugar, R. M. Shelby, C. Murray, K. Rod- bell, and J. Sleight, arXiv:2503.12514 (2025).

[7] M. P. Bland, F. Bahrami, J. G. C. Martinez, P. H. Prestegaard, B. M. Smitham, A. Joshi, E. Hedrick, A. Pakpour-Tabrizi, S. Kumar, A. Jindal, R. D. Chang, A. Yang, G. Cheng, N. Yao, R. J. Cava, N. P. de Leon, and A. A. Houck, arXiv:2503.14798 (2025).

[8] C. R. H. McRae, H. Wang, J. Gao, M. R. Vissers, T. Brecht, A. Dunsworth, D. P. Pappas, and J. Mutus, Review of Scientific Instruments 91, 091101 (2020).

[9] N. P. De Leon, K. M. Itoh, D. Kim, K. K. Mehta, T. E. Northup, H. Paik, B. Palmer, N. Samarth, S. Sangtawesin, and D. W. Steuerman, Science 372, eabb2823 (2021).

[10] I. Siddiqi, Nature Reviews Materials 6, 875 (2021).

[11] C. Müller, J. H. Cole, and J. Lisenfeld, Reports on Progress in Physics 82, 124501 (2019).

[12] J. Aumentado, M. W. Keller, J. M. Martinis, and M. H. Devoret, Phys. Rev. Lett. 92, 066802 (2004).

[13] P. J. de Visser, J. J. A. Baselmans, P. Diener, S. J. C. Yates, A. Endo, and T. M. Klapwijk, Phys. Rev. Lett. 106, 167004 (2011).

[14] R. Barends, J. Wenner, M. Lenander, Y. Chen, R. C. Bialczak, J. Kelly, E. Lucero, P. O'Malley, M. Mariantoni, D. Sank, H. Wang, T. C. White, Y. Yin, J. Zhao, A. N. Cleland, J. M. Martinis, and J. J. A. Baselmans, Applied Physics Letters 99, 113507 (2011).

[15] L. I. Glazman and G. Catelani, SciPost Phys. Lect. Notes, 31 (2021).

[16] F. Bahrami, M. P. Bland, N. Shumiya, R. D. Chang, E. Hedrick, R. A. McLellan, K. D. Crowley, A. Dutta, L. B.-V. Horn, Y. Iguchi, A. K. Anbalagan, G. Cheng, C. Yang, N. Yao, A. L. Walter, A. M. Barbour, S. Gopalakrishnan, R. J. Cava, A. A. Houck, and N. P. de Leon, arXiv:2503.03168 (2025).

[17] S. Kono, J. Pan, M. Chegnizadeh, X. Wang, A. Youssefi, M. Scigliuzzo, and T. J. Kippenberg, Nature Communications 15, 3950 (2024).

[18] L. B. Ioffe, V. B. Geshkenbein, C. Helm, and G. Blatter, Phys. Rev. Lett. 93, 057001 (2004).

[19] H. Zhou, E. Li, K. Godeneli, Z.-H. Zhang, S. Jahanbani, K. Yu, M. Odeh, S. Aloni, S. Griffin, and A. Sipahigil, arXiv:2409.10626 (2024).

[20] C. Yu and A. Leggett, Comments Condens. Matter Phys. 14, 231 (1988).

[21] L. Faoro and L. B. Ioffe, Phys. Rev. B 91, 014201 (2015).

[22] L. Grünhaupt, N. Maleeva, S. T. Skacel, M. Calvo, F. Levy-Bertrand, A. V. Ustinov, H. Rotzinger, A. Mon- fardini, G. Catelani, and I. M. Pop, Phys. Rev. Lett. 121, 117001 (2018).

[23] M. Kristen, J. N. Voss, M. Wildermuth, A. Bilmes, J. Lisenfeld, H. Rotzinger, and A. V. Ustinov, Phys. Rev. Lett. 132, 217002 (2024).

[24] V. Gupta, P. Winkel, N. Thakur, P. van Vlaanderen, Y. Wang, S. Ganjam, L. Frunzio, and R. J. Schoelkopf,

Phys. Rev. Appl. 23, 054067 (2025).

[25] P. C. J. J. Coumou, M. R. Zuiddam, E. F. C. Driessen, P. J. de Visser, J. J. A. Baselmans, and T. M. Klapwijk, IEEE Transactions on Applied Superconductivity 23, 7500404 (2013).

[26] K. R. Amin, C. Ladner, G. Jourdan, S. Hentz, N. Roch, and J. Renard, Applied Physics Letters 120, 164001 (2022).

[27] T. Charpentier, D. Perconte, S. Léger, K. R. Amin, F. Blondelle, F. Gay, O. Buisson, L. Ioffe, A. Khvalyuk, I. Poboiko, M. Feigel'man, N. Roch, and B. Sacépé, Nature Physics 21, 104 (2025).

[28] T. F. Q. Larson, S. G. Jones, T. Kalmár, P. A. Sanchez, S. P. Chitta, V. Verma, K. Genter, K. Cicak, S. W. Nam, G. Fülüp, J. Koch, R. W. Simmonds, and A. Gye- nis, arXiv:2504.07950 (2025).

[29] B. Sacépé, M. Feigel'man, and T. M. Klapwijk, Nature Physics 16, 734 (2020).

[30] M. Javaheri Rahim, T. Lehleiter, D. Bothner, C. Krell- ner, D. Koelle, R. Kleiner, M. Dressel, and M. Schef- fler, Journal of Physics D: Applied Physics 49, 395501 (2016).

[31] A. Megrant, C. Neill, R. Barends, B. Chiaro, Y. Chen, L. Feigl, J. Kelly, E. Lucero, M. Mariantoni, P. J. J. O'Malley, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Y. Yin, J. Zhao, C. J. Palmström, J. M. Martinis, and A. N. Cleland, Applied Physics Letters 100, 113510 (2012).

[32] J. Verjauw, A. Potočnik, M. Mongillo, R. Acharya, F. Mohiyaddin, G. Simion, A. Pacco, T. Ivanov, D. Wan, A. Vanleenhove, L. Souriau, J. Jussot, A. Thiam, J. Swerts, X. Piao, S. Couet, M. Heyns, B. Govoreanu, and I. Radu, Physical Review Applied 16, 014018 (2021).

[33] K. D. Crowley, R. A. McLellan, A. Dutta, N. Shumiya, A. P. M. Place, X. H. Le, Y. Gang, T. Madhavan, M. P. Bland, R. Chang, N. Khedkar, Y. C. Feng, E. A. Um- barkar, X. Gui, L. V. H. Rodgers, Y. Jia, M. M. Feld- man, S. A. Lyon, M. Liu, R. J. Cava, A. A. Houck, and N. P. de Leon, Phys. Rev. X 13, 041005 (2023).

[34] A. Romanenko, R. Pilipenko, S. Zorzetti, D. Frolov, M. Awida, S. Belomestnykh, S. Posen, and A. Gras- sellino, Phys. Rev. Appl. 13, 034032 (2020).

[35] D. C. Mattis and J. Bardeen, Phys. Rev. 111, 412 (1958).

[36] A. A. Abrikosov and L. P. Gor'kov, Soviet Physics JETP 8, 1090 (1959).

[37] A. B. Pippard, Proceedings of the Royal Society of Lon- don. Series A. Mathematical and Physical Sciences 216, 547 (1953).

[38] H. Wang, M. Hofheinz, J. Wenner, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O'Connell, D. Sank, M. Weides, A. N. Cleland, and J. M. Martinis, Applied Physics Letters 95, 233508 (2009).

[39] C. Wang, C. Axline, Y. Y. Gao, T. Brecht, Y. Chu, L. Frunzio, M. H. Devoret, and R. J. Schoelkopf, Applied Physics Letters 107, 162601 (2015).

[40] G. Calusine, A. Melville, W. Woods, R. Das, C. Stull, V. Bolkhovsky, D. Braje, D. Hover, D. K. Kim, X. Miloshi, D. Rosenberg, A. Sevi, J. L. Yoder, E. Dauler, and W. D. Oliver, Applied Physics Letters 112, 062601 (2018).
[41] W. Woods, G. Calusine, A. Melville, A. Sevi, E. Golden, D. Kim, D. Rosenberg, J. Yoder, and W. Oliver, Phys. Rev. Appl. 12, 014012 (2019).

[42] A. Melville, G. Calusine, W. Woods, K. Serniak, E. Golden, B. M. Niedzielski, D. K. Kim, A. Sevi, J. L. Yoder, E. A. Dauler, and W. D. Oliver, Applied Physics Letters 117, 124004 (2020).

[43] J. M. Sage, V. Bolkhovsky, W. D. Oliver, B. Turek, and P. B. Welander, Journal of Applied Physics 109, 063915 (2011).

[44] T. Charpentier, *Quantum circuits and the superconductor-insulator transition in a disordered superconductor*, Ph.D. thesis, Université Grenoble Alpes (2023), Available at https://theses.hal.science/tel-04405951v1.

[45] H. Deng, Z. Song, R. Gao, T. Xia, F. Bao, X. Jiang, H.-S. Ku, Z. Li, X. Ma, J. Qin, H. Sun, C. Tang, T. Wang, F. Wu, W. Yu, G. Zhang, X. Zhang, J. Zhou, X. Zhu, Y. Shi, H.-H. Zhao, and C. Deng, Phys. Rev. Appl. 19, 024013 (2023).

[46] E. T. Mannila, P. Samuelsson, S. Simbierowicz, J. T. Peltonen, V. Vesterinen, L. Grönberg, J. Hassel, V. F. Maisi, and J. P. Pekola, Nature Physics 18, 145 (2021).

[47] A. V. Semenov, I. A. Devyatov, P. J. de Visser, and T. M. Klapwijk, Phys. Rev. Lett. 117, 047002 (2016).

[48] G. Catelani and D. M. Basko, SciPost Phys. 6, 013 (2019).

[49] P. Fischer and G. Catelani, Phys. Rev. Appl. 19, 054087 (2023).

[50] C. H. Liu, D. C. Harrison, S. Patel, C. D. Wilen, O. Rafferty, A. Shearrow, A. Ballard, V. Iaia, J. Ku, B. L. T. Plourde, and R. McDermott, Phys. Rev. Lett. 132, 017001 (2024).

[51] P. B. Fischer and G. Catelani, SciPost Phys. 17, 070 (2024).

[52] R. Benevides, M. Drimmer, G. Bisson, F. Adinolfi, U. v. Lüpke, H. M. Doeleman, G. Catelani, and Y. Chu, Phys. Rev. Lett. 133, 060602 (2024).

[53] S. A. H. de Rooij, J. J. A. Baselmans, J. Bueno, V. Murugesan, D. J. Thoen, and P. J. de Visser, arXiv:2503.07443 (2025).

[54] E. Yelton, C. P. Larson, K. Dodge, K. Okubo, and B. L. T. Plourde, arXiv:2503.09554 (2025).

[55] K. Karatsu, A. Endo, J. Bueno, P. J. de Visser, R. Barends, D. J. Thoen, V. Murugesan, N. Tomita, and J. J. A. Baselmans, Applied Physics Letters 114, 032601 (2019).

[56] A. P. Vepsäläinen, A. H. Karamlou, J. L. Orrell, A. S. Dogra, B. Loer, F. Vasconcelos, D. K. Kim, A. J. Melville, B. M. Niedzielski, J. L. Yoder, S. Gustavsson, J. A. Formaggio, B. A. VanDevender, and W. D. Oliver, Nature 584, 551 (2020).

[57] C. D. Wilen, S. Abdullah, N. A. Kurinsky, C. Stanford, L. Cardani, G. D’Imperio, C. Tomei, L. Faoro, L. B. Ioffe, C. H. Liu, A. Opremcak, B. G. Christensen, J. L. DuBois, and R. McDermott, Nature 594, 369 (2021).

[58] M. McEwen, L. Faoro, K. Arya, A. Dunsworth, T. Huang, S. Kim, B. Burkett, A. Fowler, F. Arute, J. C. Bardin, A. Bengtsson, A. Bilmes, B. B. Buckley, N. Bushnell, Z. Chen, R. Collins, S. Demura, A. R. Derk, C. Erickson, M. Giustina, S. D. Harrington, S. Hong, E. Jeffrey, J. Kelly, P. V. Klimov, F. Kostritsa, P. Laptev, A. Locharla, X. Mi, K. C. Miao, S. Montazeri, J. Mutus, O. Naaman, M. Neeley, C. Neill, A. Opremcak, C. Quintana, N. Redd, P. Roushan, D. Sank, K. J. Satzinger, V. Shvarts, T. White, Z. J. Yao, P. Yeh, J. Yoo, Y. Chen, V. Smelyanskiy, J. M. Martinis, H. Neven, A. Megrant, L. Ioffe, and R. Barends, Nature Physics 18, 107 (2021).

[59] L. Cardani, I. Colantoni, A. Cruciani, F. D. Dimicis, G. D’Imperio, M. Laubenstein, A. Mariani, L. Pagnanini, S. Pirro, C. Tomei, N. Casali, F. Ferroni, D. Frolov, L. Gironi, A. Grassellino, M. Junker, C. Kopas, E. Lachman, C. R. H. McRae, J. Mutus, M. Nastasi, D. P. Pappas, R. Pilipenko, M. Sisti, V. Pettinacci, A. Romanenko, D. V. Zanten, M. Vignati, J. D. Withrow, and N. Z. Zhelev, European Physical Journal C 83, 94 (2023).

[60] M. Kharitonov, T. Proslier, A. Glatz, and M. J. Pellin, Phys. Rev. B 86, 024514 (2012).

[61] Y. V. Fominov, M. Houzet, and L. I. Glazman, Phys. Rev. B 84, 224517 (2011).

[62] A. Larkin and Y. Ovchinnikov, Sov. Phys. JETP 34, 1144 (1972).

[63] M. V. Feigel’man and M. A. Skvortsov, Phys. Rev. Lett. 109, 147002 (2012).

[64] M. A. Skvortsov and M. V. Feigel’man, J. Exp. Theor. Phys. 117, 487 (2013).

[65] B. Sacépé, C. Chapelier, T. I. Baturina, V. M. Vinokur, M. R. Baklanov, and M. Sanquer, Phys. Rev. Lett. 101, 157006 (2008).

[66] M. Chand, G. Saraswat, A. Kamlapure, M. Mondal, S. Kumar, J. Jesudasan, V. Bagwe, L. Benfatto, V. Tripathi, and P. Raychaudhuri, Phys. Rev. B 85, 014508 (2012).

[67] B. Sacépé, T. Dubouchet, C. Chapelier, M. Sanquer, M. Ovadia, D. Shahar, M. Feigel’man, and L. Ioffe, Nature Physics 7, 239 (2011).

[68] S. Mandal, S. Dutta, S. Basistha, I. Roy, J. Jesudasan, V. Bagwe, L. Benfatto, A. Thamizhavel, and P. Raychaudhuri, Phys. Rev. B 102, 060501 (2020).

[69] A. Bespalov, M. Houzet, J. S. Meyer, and Y. V. Nazarov, Phys. Rev. Lett. 117, 117002 (2016).

[70] S. De Graaf, L. Faoro, L. Ioffe, S. Mahashabde, J. Burnett, T. Lindström, S. Kubatkin, A. Danilov, and A. Y. Tzalenchuk, Science advances 6, eabc5055 (2020).

[71] C. Wang, Y. Y. Gao, I. M. Pop, U. Vool, C. Axline, T. Brecht, R. W. Heeres, L. Frunzio, M. H. Devoret, G. Catelani, L. I. Glazman, and R. J. Schoelkopf, Nature Communications 5, 5836 (2014).

[72] P. de Visser, D. Goldie, P. Diener, S. Withington, J. Baselmans, and T. Klapwijk, Physical Review Letters 112, 047004 (2014).

[73] S. A. H. de Rooij, R. Fermin, K. Kouwenhoven, T. Cop pens, V. Murugesan, D. J. Thoen, J. Aarts, J. J. A. Baselmans, and P. J. de Visser, arXiv:2410.18802 (2024).

[74] A. A. Abrikosov and L. P. Gor’kov, Zhur. Eksptl’. i Teoret. Fiz. 39 (1960).

[75] M. Checchin, D. Frolov, A. Lunin, A. Grassellino, and A. Romanenko, Physical Review Applied 18, 034013 (2022).

[76] Z.-H. Zhang, K. Godeneli, J. He, M. Odeh, H. Zhou, S. Meesala, and A. Sipahigil, Phys. Rev. X 14, 041022 (2024).

[77] A. P. Read, B. J. Chapman, C. U. Lei, J. C. Curtis, S. Ganjam, L. Krayzman, L. Frunzio, and R. J. Schoelkopf, Physical Review Applied 19, 034064 (2023).
[78] P. Winkel, K. Borisov, L. Grünhaupt, D. Rieger, M. Spiecker, F. Valenti, A. V. Ustinov, W. Wernsdorfer, and I. M. Pop, Phys. Rev. X **10**, 031032 (2020).

[79] Y. Schön, J. N. Voss, M. Wildermuth, A. Schneider, S. T. Skacel, M. P. Weides, J. H. Cole, H. Rotzinger, and A. V. Ustinov, npj Quantum Materials **5**, 18 (2020).

[80] A. P. M. Place, L. V. H. Rodgers, P. Mundada, B. M. Smitham, M. Fitzpatrick, Z. Leng, A. Premkumar, J. Bryon, A. Vrajitoarea, S. Sussman, G. Cheng, T. Madhavan, H. K. Babla, X. H. Le, Y. Gang, B. Jäck, A. Gyenis, N. Yao, R. J. Cava, N. P. de Leon, and A. A. Houck, Nature Communications **12**, 1779 (2021).

[81] C. Wang, X. Li, H. Xu, Z. Li, J. Wang, Z. Yang, Z. Mi, X. Liang, T. Su, C. Yang, G. Wang, W. Wang, Y. Li, M. Chen, C. Li, K. Linghu, J. Han, Y. Zhang, Y. Feng, Y. Song, T. Ma, J. Zhang, R. Wang, P. Zhao, W. Liu, G. Xue, Y. Jin, and H. Yu, npj Quantum Information **8**, 3 (2022).

[82] K. Serniak, S. Diamond, M. Hays, V. Fatemi, S. Shankar, L. Frunzio, R. Schoelkopf, and M. Devoret, Phys. Rev. Appl. **12**, 014052 (2019).

[83] E. I. Malevannaya, V. I. Polozov, A. I. Ivanov, A. R. Matanin, N. S. Smirnov, V. V. Echeistov, D. O. Moskalev, D. A. Mikhalin, D. E. Shirokov, Y. V. Panfilov, I. A. Ryzhikov, A. V. Andriyash, and I. A. Rodionov, arXiv:2505.14274 (2025).

[84] G. Marchegiani, L. Amico, and G. Catelani, PRX Quantum **3**, 040338 (2022).

[85] X. Pan, Y. Zhou, H. Yuan, L. Nie, W. Wei, L. Zhang, J. Li, S. Liu, Z. H. Jiang, G. Catelani, L. Hu, F. Yan, and D. Yu, Nature Communications **13**, 7196 (2022).

[86] M. McEwen, K. C. Miao, J. Atalaya, A. Bilmes, A. Crook, J. Bovaird, J. M. Kreikebaum, N. Zobrist, E. Jeffrey, B. Ying, A. Bengtsson, H.-S. Chang, A. Dunsworth, J. Kelly, Y. Zhang, E. Forati, R. Acharya, J. Iveland, W. Liu, S. Kim, B. Burkett, A. Megrant, Y. Chen, C. Neill, D. Sank, M. Devoret, and A. Opremcak, arXiv:2402.15644 (2024).

[87] H. Nho, T. Connolly, P. D. Kurilovich, S. Diamond, C. G. L. Bottcher, L. I. Glazman, and M. H. Devoret, arXiv:2505.08104 (2025).

[88] A. d'Addabbo, *Applications of Kinetic Inductance Detectors to Astronomy and Particle Physics*, Ph.D. thesis, Université Grenoble Alpes (2014), Available at https://theses.hal.science/tel-01368000.

[89] A. Monfardini, J. Baselmans, A. Benoit, A. Bideaud, O. Bourrion, A. Catalano, M. Calvo, A. D'Addabbo, S. Doyle, J. Goupy, H. L. Sueur, and J. Macias-Perez, in *Millimeter, Submillimeter, and Far-Infrared Detectors and Instrumentation for Astronomy VIII*, Vol. 9914, edited by W. S. Holland and J. Zmuidzinas, International Society for Optics and Photonics (SPIE, 2016) p. 99140N.

[90] F. Henriques, F. Valenti, T. Charpentier, M. Lagoin, C. Gouriou, M. Martinez, L. Cardani, M. Vignati, L. Grünhaupt, D. Gusenkova, J. Ferrero, S. T. Skacel, W. Wernsdorfer, A. V. Ustinov, G. Catelani, O. Sander, and I. M. Pop, Applied Physics Letters **115**, 212601 (2019).

[91] M. Mohseni, A. Scherer, K. G. Johnson, O. Wertheim, M. Otten, N. A. Aadit, Y. Alexeev, K. M. Bresniker, K. Y. Camsari, B. Chapman, S. Chatterjee, G. A. Dagnew, A. Esposito, F. Fahim, M. Fiorentino, A. Gajjar, A. Khalid, X. Kong, B. Kulchytskyy, E. Kyoseva, R. Li, P. A. Lott, I. L. Markov, R. F. McDermott, G. Pedretti, P. Rao, E. Rieffel, A. Silva, J. Sorebo, P. Spentzouris, Z. Steiner, B. Torosov, D. Venturelli, R. J. Visser, Z. Webb, X. Zhan, Y. Cohen, P. Ronagh, A. Ho, R. G. Beausoleil, and J. M. Martinis, arXiv:2411.10406 (2025).

[92] Google Quantum AI and Collaborators, Nature **638**, 920 (2024).

[93] B. Pashinsky, M. V. Feigelman, and A. Andreev, SciPost Phys. **14**, 10.21468/SciPostPhys.14.5.096 (2023).

[94] C. J. K. Richardson, A. Alexander, C. G. Weddle, B. Arey, and M. Olszta, Journal of Applied Physics **127**, 235302 (2020).

[95] A. Kamenev, Field theory of non-equilibrium systems (Cambridge University Press, 2023).

[96] A. Schmid and G. Schön, Journal of Low Temperature Physics **20**, 207 (1975).

[97] R. Gao, W. Yu, H. Deng, H.-S. Ku, Z. Li, M. Wang, X. Miao, Y. Lin, and C. Deng, Physical Review Materials **6**, 036202 (2022).

[98] A. Shnirman, Y. Makhlin, and G. Schön, Physica Scripta **2002**, 147 (2002).

[99] G. Catelani and D. M. Basko, SciPost Physics **6**, 013 (2019).

[100] J. S. Meyer, private communications.

[101] N. Mott, Philosophical Magazine **22**, 7 (1970).

[102] Paper [93] contains numerical mistake: factor $8\pi$ in eq.(2) and few similar equations should actually be replaced by factor $4\pi$; we used this corrected version here.

[103] G. Stan, S. B. Field, and J. M. Martinis, Phys. Rev. Lett. **92**, 097003 (2004).

[104] J. Burnett, A. Bengtsson, D. Niepce, and J. Bylander, Journal of Physics: Conference Series **969**, 012131 (2018).

[105] D. Lopez-Nunez, Q. P. Montserrat, G. Rius, E. Bertoldo, A. Torras-Coloma, M. Martinez, and P. Forn-Diaz, arXiv:2311.14119 (2023).

[106] C. T. Earnest, J. H. Béjanin, T. G. McConkey, E. A. Peters, A. Korinek, H. Yuan, and M. Mariantoni, Superconductor Science and Technology **31**, 125013 (2018).

[107] C. J. K. Richardson, N. P. Siwak, J. Hackley, Z. K. Keane, J. E. Robinson, B. Arey, I. Arslan, and B. S. Palmer, Superconductor Science and Technology **29**, 064003 (2016).

[108] J. Biznarova, A. Osman, E. Rehnman, L. Chayanun, C. Krizan, P. Malmberg, M. Rommel, C. Warren, P. Delsing, A. Yurgens, J. Bylander, and A. Fadavi Roudsari, npj Quantum Information **10**, 78 (2024).

[109] C. Kurter, C. E. Murray, R. T. Gordon, B. B. Wymore, M. Sandberg, R. M. Shelby, A. Eddins, V. P. Adiga, A. D. K. Finck, E. Rivera, A. A. Stabile, B. Trimm, B. Wacaser, K. Balakrishnan, A. Pyzyna, J. Sleight, M. Steffen, and K. Rodbell, npj Quantum Information **8**, 31 (2022).

[110] L. Krayzman, C. U. Lei, S. Ganjam, J. Teoh, L. Frunzio, and R. J. Schoelkopf, Applied Physics Letters **124**, 204001 (2024).

[111] M. Reagor, H. Paik, G. Catelani, L. Sun, C. Axline, E. Holland, I. M. Pop, N. A. Masluk, T. Brecht, L. Frunzio, M. H. Devoret, L. Glazman, and R. J. Schoelkopf, Applied Physics Letters **102**, 192604 (2013).

[112] M. Kudra, J. Biznarova, A. Fadavi Roudsari, J. J. Burnett, D. Niepce, S. Gasparinetti, B. Wickman, and P. Delsing, Applied Physics Letters **117**, 070601 (2020).
[113] S. Chakram, A. E. Oriani, R. K. Naik, A. V. Dixit, K. He, A. Agrawal, H. Kwon, and D. I. Schuster, Phys. Rev. Lett. 127, 107701 (2021).

[114] J. C. Esmenda, E. A. Laird, I. Bailey, N. Du, S. Durham, G. Carosi, T. Gamble, P. Smith, E. Daw, and Y. A. Pashkin, arXiv:2503.22637 (2025).

[115] H. Rotzinger, S. T. Skacel, M. Pfirrmann, J. N. Voss, J. Münzberg, S. Probst, P. Bushev, M. P. Weides, A. V. Ustinov, and J. E. Mooij, Superconductor Science and Technology 30, 025002 (2016).

[116] W. Zhang, K. Kalashnikov, W.-S. Lu, P. Kamenov, T. DiNapoli, and M. Gershenson, Physical Review Applied 11, 011003 (2019).

[117] Q. He, P. OuYang, M. Dai, H. Guan, J. Hu, S. He, Y. Wang, and L. F. Wei, AIP Advances 11, 065204 (2021).

[118] D. Rieger, S. Günzler, M. Spiecker, A. Nambisan, W. Wernsdorfer, and I. Pop, Physical Review Applied 20, 014059 (2023).

[119] S. Günzler, D. Rieger, and I. Pop, Private communication (2024).

[120] C. Roy, S. Frasca, and P. Scarlino, arXiv:2503.13321 (2025).

[121] M. Khorramshahi, M. Spiecker, P. Paluch, S. Geisert, N. Gosling, N. Zapata, L. Brauch, C. Kuebel, S. Dehm, R. Krupke, W. Wernsdorfer, I. M. Pop, and T. Reisinger, arXiv:2506.07761 (2025).

[122] G. Coiffard, M. Daal, N. Zobrist, N. Swimmer, S. Steiger, B. Bumble, and B. A. Mazin, Superconductor Science and Technology 33, 07LT02 (2020).

[123] X. Li, A. Suzuki, and M. Garcia-Sciveres, arXiv:2502.19818 (2025).

[124] S. Zhu, F. Crisa, M. Bal, A. A. Murthy, J. Lee, Z. Sung, A. Lunin, D. Frolov, R. Pilipenko, D. Bafia, A. Mitra, A. Romanenko, and A. Grassellino, arXiv:2207.13024 (2022).

[125] M. V. P. Altoé, A. Banerjee, C. Berk, A. Hajr, A. Schwartzberg, C. Song, M. Alghadeer, S. Aloni, M. J. Elowson, J. M. Kreikebaum, E. K. Wong, S. M. Griffin, S. Rao, A. Weber-Bargioni, A. M. Minor, D. I. Santiago, S. Cabrini, I. Siddiqi, and D. F. Ogletree, PRX Quantum 3, 020312 (2022).

[126] T. Noguchi, A. Dominjon, M. Kroug, S. Mima, and C. Otani, IEEE Transactions on Applied Superconductivity 29, 1 (2019).

[127] C. Varmazis and M. Strongin, Physical Review B 10, 1885 (1974).

[128] H. Wang, T. Banerjee, T. G. Farinha, A. T. Hanbicki, V. Fatemi, B. S. Palmer, and C. J. K. Richardson, arXiv:2402.18051 (2024).

[129] M. Drimmer, S. Telkamp, F. L. Fischer, I. C. Rodrigues, C. Todt, F. Krizek, D. Kriegner, C. Müller, W. Wegscheider, and Y. Chu, arXiv:2403.12164 (2024).

[130] M. de Ory, D. Rodriguez, M. Magaz, V. Rollano, D. Granados, and A. Gomez, arXiv:2401.14764 (2024).

[131] M. Bal, A. A. Murthy, S. Zhu, F. Crisa, X. You, Z. Huang, T. Roy, J. Lee, D. V. Zanten, R. Pilipenko, I. Nekrashevich, A. Lunin, D. Bafia, Y. Krasnikova, C. J. Kopas, E. O. Lachman, D. Miller, J. Y. Mutus, M. J. Reagor, H. Cansizoglu, J. Marshall, D. P. Pappas, K. Vu, K. Yadavalli, J.-S. Oh, L. Zhou, M. J. Kramer, F. Lecocq, D. P. Goronzy, C. G. Torres-Castanedo, P. G. Pritchard, V. P. Dravid, J. M. Rondinelli, M. J. Bedzyk, M. C. Hersam, J. Zasadzinski, J. Koch, J. A. Sauls, A. Romanenko, and A. Grassellino, npj Quantum Information 10, 43 (2024).

[132] B. Abdisatarov, D. Bafia, A. Murthy, G. Eremeev, H. E. Elsayed-Ali, J. Lee, A. Netepenko, C. P. A. Carlos, S. Leith, G. J. Rosaz, A. Romanenko, and A. Grassellino, Applied Physics Letters 125, 124002 (2024).

[133] A. Romanenko, R. Pilipenko, S. Zorzetti, D. Frolov, M. Awida, S. Belomestnykh, S. Posen, and A. Grassellino, Physical Review Applied 13, 034032 (2020).

[134] A. E. Oriani, Multimodal and Ultra High-Q Superconducting Niobium Cavities for Circuit Quantum Electrodynamics, Ph.D. thesis, The University of Chicago (2022).

[135] P. Heidler, C. M. F. Schneider, K. Kustura, C. Gonzalez-Ballestero, O. Romero-Isart, and G. Kirchmair, Physical Review Applied 16, 034024 (2021).

[136] O. Milul, B. Guttel, U. Goldblatt, S. Hazanov, L. M. Joshi, D. Chausovsky, N. Kahn, E. Çiftyürek, F. Lafont, and S. Rosenblum, PRX Quantum 4, 030336 (2023).

[137] A. E. Oriani, F. Zhao, T. Roy, A. Anferov, K. He, A. Agrawal, R. Banerjee, S. Chakram, and D. I. Schuster, arXiv:2403.00286 (2024).

[138] A. Suleymanzade, A. Anferov, M. Stone, R. K. Naik, A. Oriani, J. Simon, and D. Schuster, Applied Physics Letters 116, 104001 (2020).

[139] S. Frasca, I. Arabadzhiev, S. B. de Puechredon, F. Oppliger, V. Jouanny, R. Musio, M. Scigliuzzo, F. Minganti, P. Scarlino, and E. Charbon, Physical Review Applied 20, 044021 (2023).

[140] D. Niepce, J. Burnett, and J. Bylander, Physical Review Applied 11, 044014 (2019).

[141] P. Foshat, P. Baity, S. Danilin, V. Seferai, S. Poorgholam-Khanjari, H. Feng, O. A. Mukhanov, M. Hutchings, R. H. Hadfield, M. Imran, M. Weides, and K. Delfanazari, arXiv:2306.02356 (2023).

[142] C. X. Yu, S. Zihlmann, G. T. Fernández-Bada, J.-L. Thomassin, F. Gustavo, É. Dumur, and R. Maurand, Applied Physics Letters 118, 054001 (2021).

[143] X.-Y. Wei, J.-Z. Pan, Y.-P. Lu, J.-L. Jiang, Z.-S. Li, S. Lu, X.-C. Tu, Q.-Y. Zhao, X.-Q. Jia, L. Kang, J. Chen, C.-H. Cao, H.-B. Wang, W.-W. Xu, G.-Z. Sun, and P.-H. Wu, Chinese Physics B 29, 128401 (2020).

[144] X. Wei, J. Jiang, W. Xu, T. Guo, K. Zhang, Z. Li, T. Zhou, Y. Sheng, C. Cao, G. Sun, and P. Wu, Applied Physics Letters 123, 154005 (2023).

[145] H. I. Sueur, A. Svilans, N. Bourlet, A. Murani, L. Bergé, L. Dumoulin, and P. Joyez, arXiv:1810.12801 (2018).

[146] N. Samkharadze, A. Bruno, P. Scarlino, G. Zheng, D. DiVincenzo, L. DiCarlo, and L. Vandersypen, Physical Review Applied 5, 044004 (2016).

[147] A. Bruno, G. de Lange, S. Asaad, K. L. van der Enden, N. K. Langford, and L. DiCarlo, Applied Physics Letters 106, 182601 (2015).

[148] A. Giachero, M. R. Vissers, J. D. Wheeler, M. Malnou, J. E. Austermann, J. Hubmayr, A. Nucciotti, J. N. Ulom, and J. Gao, IEEE Transactions on Applied Superconductivity 33, 1 (2023).

[149] M. Yang, X. He, W. Gao, J. Chen, Y. Wu, X. Wang, G. Mu, W. Peng, and Z. Lin, AIP Advances 14, 085027 (2024).

[150] I. P. C. Cools, R. M. López-Báez, V. Buccheri, O. Shvetsov, N. Trnjanin, E. Hogedal, and S. P. Dash, Journal of Physics D: Applied Physics 58, 255102 (2024).
(2025).

[151] P. Szypryt, B. A. Mazin, G. Ulbricht, B. Bumble, S. R. Meeker, C. Bockstiegel, and A. B. Walter, *Applied Physics Letters* **109**, 151102 (2016).

[152] E. Dumur, B. Delsol, T. Weissl, B. Kung, W. Guichard, C. Hoarau, C. Naud, K. Hasselbach, O. Buisson, K. Ratter, and B. Gilles, *IEEE Transactions on Applied Superconductivity* **26**, 1 (2016).

[153] D. P. Lozano, M. Mongillo, X. Piao, S. Couet, D. Wan, Y. Canvel, A. M. Vadiraj, T. Ivanov, J. Verjauw, R. Acharya, J. V. Damme, F. A. Mohiyaddin, J. Jussoot, P. P. Gowda, A. Pacco, B. Raes, J. V. de Vondel, I. P. Radu, B. Govoreanu, J. Swerts, A. Potocnik, and K. D. Greve, *Materials for Quantum Technology* **4**, 025801 (2024).

[154] H. Jia, B. Zhou, T. Wang, Y. Wu, L. Yang, Z. Ding, S. Li, X. Cai, K. Xiong, and J. Feng, *Journal of Vacuum Science & Technology B* **41**, 052210 (2023).

[155] L. Shi, T. Guo, R. Su, T. Chi, Y. Sheng, J. Jiang, C. Cao, J. Wu, X. Tu, G. Sun, J. Chen, and P. Wu, *Applied Physics Letters* **121**, 242601 (2022).

[156] T. A. van Schijndel, A. P. McFadden, A. N. Engel, J. T. Dong, W. J. Yánez-Parreno, M. Parthasarathy, R. W. Simmonds, and C. J. Palmström, *Physical Review Applied* **23**, 034025 (2025).

[157] D. Li, J. Hu, Y. Li, and S. An, arXiv:2405.02955 (2024).

[158] G. Marcaud, D. Perello, C. Chen, E. Umbarkar, C. Weiland, J. Gao, S. Diez, V. Ly, N. Mahuli, N. D'Souza, Y. He, S. Aghaeimeibodi, R. Resnick, C. Jaye, A. K. Rumaiz, D. A. Fischer, M. Hunt, O. Painter, and I. Jarige, arXiv:2501.09885 (2025).

[159] K. Kouwenhoven, D. Fan, E. Biancalani, S. A. de Rooij, T. Karim, C. S. Smith, V. Murugesan, D. J. Thoen, J. J. Baselmans, and P. J. de Visser, *Physical Review Applied* **19**, 034007 (2023).

[160] E. T. Holland, Y. J. Rosen, N. Materise, N. Woollett, T. Voisin, Y. M. Wang, S. G. Torres, J. Mireles, G. Carosi, and J. L. DuBois, *Applied Physics Letters* **111**, 202602 (2017).

[161] P. R. Carriere, P. Frigola, R. Agustsson, M. H. Chou, A. N. Cleland, T. Horn, M. Kelly, S. V. Kutsaev, R. Povey, and A. Y. Smirnov, *IOP Conference Series: Materials Science and Engineering* **1241**, 012046 (2022).

[162] A. Shearrow, G. Koolstra, S. J. Whiteley, N. Earnest, P. S. Barry, F. J. Heremans, D. D. Awschalom, E. Shirkooff, and D. I. Schuster, *Applied Physics Letters* **113**, 212601 (2018).

[163] R. Mencia, Y.-H. Lin, and V. Manucharyan, *Journal of Applied Physics* **130**, 225109 (2021).

[164] S. Ohya, B. Chiaro, A. Megrant, C. Neill, R. Barends, Y. Chen, J. Kelly, D. Low, J. Mutus, P. J. J. O'Malley, P. Roushan, D. Sank, A. Vainsencher, J. Wenner, T. C. White, Y. Yin, B. D. Schultz, C. J. Palmström, B. A. Mazin, A. N. Cleland, and J. M. Martinis, *Superconductor Science and Technology* **27**, 015009 (2013).

[165] G. Koolstra, E. Glen, N. Beysengulov, H. Byeon, K. Castoria, M. Sammon, B. Dizdar, C. Wang, D. Schuster, S. Lyon, J. Pollanen, and D. Rees, *Phys. Rev. Appl.* **23**, 024001 (2025).

[166] Y. Wu, Y. Liu, W. Gao, X. Wang, X. He, H. Jin, W. Peng, G. Mu, Z. Lin, and Z. Wang, *Superconductor Science and Technology* **37**, 115002 (2024).

[167] R. Gao, H. Ku, H. Deng, W. Yu, T. Xia, F. Wu, Z. Song, M. Wang, X. Miao, C. Zhang, Y. Lin, Y. Shi, H. Zhao, and C. Deng, *Advanced Materials* **34**, 2201268 (2022).

[168] R. Gao, Private communications.

[169] O. Quaranta, T. W. Cecil, and A. Miceli, *IEEE Transactions on Applied Superconductivity* **23**, 2400104 (2013).

[170] N. Kirsh, E. Svetitsky, S. Goldstein, G. Pardo, O. Hachmo, and N. Katz, *Physical Review Applied* **16**, 044017 (2021).
# SUPPLEMENTARY MATERIALS

## A. Complex conductivity

Here, we introduce the key quantities that describe the electrodynamics of superconductors. In the London gauge, the London equation $\mathbf{j}_s = \omega\sigma_2\mathbf{A}$ relates the superconducting current density $\mathbf{j}_s$ to the vector potential $\mathbf{A}$ where the proportionality constant $\sigma_2$ is the imaginary part of the complex conductivity $\sigma(\omega)$. A direct consequence of this equation is that magnetic fields can enter a superconductor only over the magnetic penetration depth $\lambda = 1/\sqrt{\mu_0\omega\sigma_2}$. The surface impedance of a superconductor $Z_s = 1/\lambda\sigma(\omega)$ consists mainly in a dissipationless inductive channel, with characteristic inductance $L_K = 1/\omega\lambda\sigma_2 = \mu_0\lambda$, called the kinetic inductance, that describes the kinetic energy stored by the Cooper pair condensate. For thin films of thickness $d < \lambda$, the magnetic field penetrates entirely and $L_K = 1/\omega d\sigma_2 = \mu_0\lambda^2/d$.

## B. Microwave loss due to nonequilibrium quasiparticles in clean and moderately disordered superconductors

In this section, we describe the possible loss mechanisms arising from nonequilibrium quasiparticles, with a focus on distinguishing between localized and delocalized quasiparticles. The specific dissipation mechanism associated with these quasiparticles depends on a combination of material properties, experimental conditions, and the characteristics of the external quasiparticle source. These factors conspire to establish a steady state of the quasiparticle subsystem. Remarkably, the most relevant regimes can be classified based on a small set of physical parameters.

Subsections B1 and B2 introduce the key features of the quasiparticle density of states and distribution function, respectively. Subsection B3 provides a qualitative overview of dissipation mechanisms, classifying them by

their relative dissipation strength (see also Supplementary Materials Fig. 3). The subsequent subsections, B4–B8, examine each mechanism in detail and, where possible, provide estimates for the associated quality factor. We also briefly review theoretical models describing the power dependence of the quality factor, as this behavior further accentuates the qualitative differences between dissipation mechanisms.

Although we have aimed to provide a comprehensive overview of the relevant physical mechanisms, the crude nature of some of our estimates –or the absence of estimates in certain regimes– combined with the empirical trends observed in Fig. 1, should serve as a strong motivation for further theoretical and experimental investigations into the kinetics of localized quasiparticles.

### 1. Quasiparticle Density of States

One of the key quantities for understanding quasiparticle-induced dissipation is the quasiparticle density of states per spin projection, $\nu(E)$. It is well known [62, 64, 70] that $\nu(E)$ in a conventional superconductor – either clean ($l \gg \xi_0$) or moderately disordered ($\xi_0 \gg l \gg \lambda_F$, with $\lambda_F$ the Fermi wavelength) – consists of two components (see Supplementary Materials Fig. 2a):

i) A sharp BCS peak, $\nu_{\text{bulk}}(E)$, associated with the spectrum of delocalized quasiparticle states. This part of the density of states ends at the mobility edge $E_g$, which lies slightly below the bulk superconducting order parameter $\Delta_0$. Both the difference $\Delta_0 - E_g$ and the finite broadening of the BCS peak originate from nonmagnetic impurity scattering [62].

ii) A rapidly decaying tail of localized states $\nu_{\text{loc}}(E < E_g)$, arising from rare spatial regions where disorder suppresses the superconducting order parameter locally, forming shallow potential wells with bound states that can trap quasiparticles. The characteristic spatial extent $r_{\text{loc}}(E)$ of such states and their corresponding density of states $\nu_{\text{loc}}(E)$ are given, for the clean ($\xi_0 \ll l$) and dirty ($\xi_0 \gg l$) limits respectively [69, 70], by:

$$ \nu_{\text{loc}}^{\text{clean}}(\varepsilon) \propto \exp\left[-\left(\frac{\varepsilon}{\varepsilon_T}\right)^{5/4}\right], \quad \frac{r_{\text{loc}}^{\text{clean}}(\varepsilon)}{2\xi_0} \sim \left(\frac{\Delta_0}{\varepsilon}\right)^{1/2}, \qquad (4) $$

$$ \nu_{\text{loc}}^{\text{dirty}}(\varepsilon) \propto \left(\frac{\varepsilon}{\varepsilon_T}\right)^{9/8} \exp\left[-\left(\frac{\varepsilon}{\varepsilon_T}\right)^{5/4}\right], \quad \frac{r_{\text{loc}}^{\text{dirty}}(\varepsilon)}{2\xi_0} = \left(\frac{2\Delta_0}{3\varepsilon}\right)^{1/4}, \qquad (5) $$

where $\varepsilon = E_g - E \ll \Delta_0$ is the energy distance to the mobility edge, and $\varepsilon_T$ characterizes the energy scale of the tail, which depends on the degree of disorder and other material parameters.

Associated with the localized part of the spectrum is the maximum concentration of localized quasiparticles
that a given material can host in a steady state:

$$n_{\text{loc}}^{\text{max}} = \int_{0}^{E_g} dE \nu_{\text{loc}}(E). \qquad (6)$$

Note that the usual factor of 2 accounting for spin degeneracy is absent here. This is because two quasiparticles occupying the same localized state would recombine immediately, allowing only single occupancy of such a state.

## 2. Quasiparticle distribution function and concentration

Another important physical quantity is the steady-state quasiparticle concentration:

$$n_{\text{qp}} := 2 \int_{0}^{+\infty} dE \nu(E) f_s(E), \qquad (7)$$

where $f_s(E)$ is the steady-state distribution function of the quasiparticles. We emphasize that both $f_s$ and, consequently, $n_{\text{qp}}$, are difficult to describe theoretically, as they depend critically on the nature of the occupied states due to the complexity of the underlying kinetics.

It is reasonable to assume that delocalized quasiparticles rapidly equilibrate with phonons. In this case, $f_s$ is approximately given by a quasi-equilibrium distribution function (assuming $T_{\text{eff}} \ll \Delta_0$):

$$f_s(E > E_g) \approx \frac{x_{\text{qp}}}{1 + e^{E/T_{\text{eff}}}} \approx x_{\text{qp}} e^{-E/T_{\text{eff}}}, \qquad (8)$$

where the prefactor $x_{\text{qp}}$, determining the concentration of delocalized quasiparticles, strongly depends on both material properties (such as phonon-assisted recombination and inelastic relaxation rates) and on the nature of the external quasiparticle source (e.g., high-energy phonons from environmental radiation). Moreover, the effective temperature $T_{\text{eff}}$ may differ significantly from the base temperature $T_0$, due to insufficient thermal coupling between the phonon subsystem and the external cooling bath.

The situation is drastically different in the case where exclusively localized states are occupied. Due to the exponentially decaying spatial profile of the localized wavefunction, the recombination rate $\Gamma_r$ of such quasiparticles (e.g., via phonon emission) is strongly suppressed. Indirect experimental evidence suggests values of $\Gamma_r$ as low as ~0.1 MHz in clean NbN films [70]. This implies that even a weak external source can maintain a stationary concentration of localized quasiparticles that is many orders of magnitude higher than the thermal equilibrium concentration at the same temperature.

A qualitative model describing this scenario was proposed in Ref. [69] for the dirty limit. However, it remains valid in the clean limit, as it relies only on the localization properties of quasiparticle states. The key observation is that the recombination rate of a localized quasiparticle due to any local process (e.g., phonon emission) can

be estimated as $\Gamma_r(E) \sim \bar{\Gamma} n_{\text{qp}} r_{\text{loc}}^3(E)$, where $\bar{\Gamma}$ is the normal-state electron-phonon relaxation rate. This estimate follows from the requirement of finite spatial overlap between the wavefunctions of recombining quasiparticles. Due to the strong energy dependence of $r_{\text{loc}}(E)$, see Eqs. (4)-(5), the recombination rate $\Gamma_r(E)$ becomes much smaller than the single-particle energy relaxation rate as the quasiparticle relaxes to lower energies.

As a result, Ref. [69] demonstrated that quasiparticle relaxation effectively halts at an energy $E_C < E_g$ (see Supplementary Materials Fig. 2a). Quasiparticles with energy above $E_C$ either relax to $E_C$ or recombine before reaching it, while recombination at $E < E_C$ is negligible due to the vanishing overlap between localized states. Further relaxation to $E < E_C$ is also blocked: the quasiparticle occupies the ground state of its local well, and phonon-assisted tunneling to deeper wells with lower energy is prevented, as those states are already occupied. Recombination remains the only possible relaxation mechanism if the quasiparticles have opposite spins.

This picture is summarized by the following ansatz for the steady-state distribution function:

$$f_s(E) \approx \frac{1}{2} \theta(E_C - E). \qquad (9)$$

The prefactor 1/2 reflects the fact that each localized state can host only one quasiparticle: the presence of two quasiparticles in the same state would lead to significant spatial overlap and rapid recombination, on a timescale much shorter than $\Gamma_r^{-1}$. Equation (9) implies a vanishing effective temperature $T_{\text{eff}}$, although poor phonon thermalization may still result in a finite $T_{\text{eff}}$ due to a residual balance between phonon heating and recombination processes for $E > E_C$. A more accurate expression for $f_s(E)$ would require a detailed microscopic model of localized quasiparticle kinetics, which lies beyond the scope of this work.

The cutoff energy $E_C$ is determined by the balance between external quasiparticle generation and recombination power. However, because the steady-state quasiparticle concentration $n_{\text{qp}}$, defined in Eq. (7), is a steep function of $E_C$ (owing to the exponential tail of $\nu_{\text{loc}}(E)$, see Eqs. (4)-(5)), the value of $E_C$ is only weakly sensitive to microscopic parameters. Instead, it is essentially fixed by the condition that the localized wavefunctions do not spatially overlap at energy $E \approx E_C$:

$$V_{\text{loc}}(E_C) n_{\text{qp}} \approx V_{\text{loc}}(E_C) \int_{0}^{E_C} dE \nu_{\text{loc}}(E) \sim 1, \qquad (10) \\ V_{\text{loc}}(E) = r_{\text{loc}}^2(E) \min\{d, r_{\text{loc}}(E)\},$$

where $V_{\text{loc}}(E)$ represents the localization volume and accommodates both thin films (finite $d$) and bulk materials ($d \to \infty$). The precise numerical factor on the right-hand side depends on microscopic details such as drive strength, shape of localized states, and recombination
mechanisms, but these affect $E_C$ only logarithmically due to the steep energy dependence of $\nu_{\text{loc}}(E)$. This observation renders the ansatz in Eq. (9) self-consistent.

The crucial practical consequence of Eqs. (4), (5), and (10) is that localized quasiparticles are most likely found in bound states of characteristic size $r_{\text{loc}}(E_C)$ at a certain energy $E_C$. This leads to a universal steady-state concentration (assuming $d = \infty$ for simplicity):

$$r_{\text{loc}}(E_C) = b\xi_0 \Leftrightarrow n_{\text{qp}}\xi_0^3 \sim \frac{1}{b^3}, \quad (11)$$

where $b = 3-10$ is the characteristic size of the localized state at energy $E_C$, expressed in units of the coherence length $\xi_0$. The value of $b$ depends only weakly on the strength of the external drive or on other microscopic details [69].

This result highlights a key distinction between the kinetics of localized and delocalized quasiparticles: for delocalized quasiparticles, the steady-state concentration $n_{\text{qp}} \gg 1/\xi_0^3$ is influenced by a broader range of system-specific factors.

We conclude this subsection by noting that a direct experimental observation of the transition between localized and delocalized quasiparticle regimes remains elusive. Most experimental probes of quasiparticle density rely on their delocalized character [71], and a generalization of such techniques to localized quasiparticles has not yet been developed.

### 3. Qualitative picture of quasiparticle dissipation

Using five parameters $-T_{\text{eff}}$, $\varepsilon_T$, $n_{\text{loc}}^{\text{max}}$, $n_{\text{qp}}$, and the external frequency $\omega$ – Supplementary Materials Fig. 3 provides a classification of the main dissipation mechanisms associated with nonequilibrium quasiparticles. The primary classification criterion is whether the delocalized part of the quasiparticle spectrum is significantly occupied. This can occur in two scenarios: i) $n_{\text{qp}} > n_{\text{loc}}^{\text{max}}$, implying that localized states are insufficient to accommodate all quasiparticles, and therefore delocalized states must be populated; or ii) $T_{\text{eff}} > \varepsilon_T$, where the effective temperature is sufficient to ionize localized states, leading to delocalized quasiparticles.

In either case, the resulting dissipation is captured by a Mattis-Bardeen-type expression [35, 61],

$$\frac{\sigma_1}{\sigma_0} \sim \frac{n_{\text{qp}}}{\nu_0 \omega} \Leftrightarrow \sigma_1 \sim \frac{\Delta_0}{\omega} n_{\text{qp}} \xi_0^3 \frac{2e^2}{\hbar \xi_0}, \quad (12)$$

describing absorption by transitions in the continuous spectrum, as detailed in subsec. B 7. This mechanism provides plausible dissipation values for 3D cavities and clean films. However, it leads to overly strong dissipation when applied to disordered thin films, inconsistent with observed quality factors. We therefore conclude that in the left-hand portion of Fig. 1, dissipation must originate from localized quasiparticles.

In the localized regime, the dissipation mechanism becomes strongly frequency-dependent, as illustrated in Supplementary Materials Fig. 3. However, it seems likely that in many relevant devices the driving frequency $\omega$ is large enough to ionize localized states directly, i.e., $\omega > E_g - E_C \sim \varepsilon_T$. As a result, the dissipation is again described by Eq. (12), but with a quasiparticle density that becomes universal in units of $\xi_0^{-3}$ (see Eq. (11)). This is sufficient to explain the quality factors of the best dirty thin-film resonators in Fig. 1.

An important note concerns the relation between $\sigma_1$ and the quality factor $Q$, given by

$$Q = \frac{\sigma_2}{\alpha \sigma_1}, \quad (13)$$

where $\alpha$ is the kinetic inductance fraction $\alpha = L_K/(L_K + L_g)$ where $L_g$ is the geometric inductance. Its value depends on the device geometry and material parameters, with $\alpha \approx 1$ for dirty superconductors and $\alpha \sim 2\omega\lambda/c \ll 1$ in the clean limit, where $\lambda$ is the London penetration depth. Nonetheless, some dirty superconductors with $l \ll \xi_0$ may still have a large enough superfluid density to yield small $\alpha$, as in the TiN samples of Ref. [94] (see Supplementary Materials Table I).

As the frequency is lowered below $\omega < \varepsilon_T$, direct ionization is no longer allowed, resulting in a rapid increase in the quality factor. Residual dissipation in this regime may arise from other mechanisms illustrated in Supplementary Materials Figs. 2d-f and 3. While these mechanisms remain poorly understood, they are unlikely to yield $\sigma_1$ values larger than those associated with direct ionization.

A notable example is the excitation of localized quasiparticles between discrete bound states within a single potential well –effectively forming a two-level system (TLS)– as proposed in Ref. [70] for clean superconductors (see Supplementary Materials Fig. 2d). In subsec. B 6, we estimate the corresponding quality factor, which turns out to be much larger than that associated with direct ionization.

Other speculative loss channels are briefly discussed in subsec. B 8, see also Supplementary Materials Fig. 2d-f.

### 4. Direct ionization in the dirty limit

Following Ref. [95, Ch. 9 and 11], we briefly recall the key elements relevant in the dirty limit, $l \ll \xi_0$. The superconducting state for a given spatial profile of the order parameter $\Delta(r)$ is described by the Usadel equation for the retarded Green’s function $g^R(\omega, r)$, a matrix in Nambu space. The diagonal component, $\nu_0 g_{11}^R$, corresponds to $\langle \psi \bar{\psi} \rangle$, while the off-diagonal component, $\nu_0 g_{12}^R$, corresponds to $\langle \psi \psi \rangle$. The local density of states is given by $\nu(\omega, r) = \nu_0 \text{Re} g_{11}^R(\omega, r)$. The Green’s function satisfies the nonlinear normalization condition $g^2 = 1$, which can be conveniently parametrized using
the complex spectral angle $\theta(\omega, r)$: namely, $g_{11}^R = \cos\theta$, and $g_{12}^R = \sin\theta$. The Usadel equation is solved self-consistently along with the gap equation for $\Delta$.

The presence of short-range disorder generates an effective depairing term in the long-wavelength limit of the Usadel equation [62], and induces uncorrelated noise in the order parameter superimposed on its uniform background value. For energies in the quasiparticle continuum, these fluctuations can be neglected, allowing one to consider a spatially uniform Green's function. In this case, the Usadel equation reduces to:

$$ \cos\theta + i\sin\theta \frac{E+i0}{\Delta} = \frac{\eta}{2}\sin 2\theta, \quad (14) $$

where $\eta$ is the dimensionless depairing parameter. The resulting density of states, $\nu_{AG}(E) = \nu_0 \text{Re} \cos\theta(E)$, follows the Abrikosov-Gor'kov profile: a smeared BCS peak with a sharp gap edge located at $E_g \approx \Delta(1 - \eta^{2/3})^{3/2}$ [62, 74].

At energies below the gap, $E < E_g$, a finite density of states arises only due to rare local dips in the order parameter. The most probable configuration leading to a subgap state at energy $E = E_g - \epsilon$, with $\epsilon \ll E_g$, is found via the method of optimal fluctuations [62, 64], and has the form:

$$ \theta(\epsilon, r) = \frac{\pi}{2} + \sqrt{\frac{\epsilon}{E_g}} \phi \left( \frac{r}{r_{\text{loc}}(\epsilon)} \cdot \frac{1}{1 - \eta^{2/3}} \right), \quad (15) $$

where $\phi \sim 1$ is a real-valued function that depends on the dimensionality of the system, and $r_{\text{loc}}(\epsilon)$ is defined in Eq. 5. Taking into account the statistical weight of such fluctuations leads to the subgap density of states $\nu_{\text{loc}}(E)$ given by Eq. (5) in the dirty limit.

The real part of the conductivity due to transitions into the quasiparticle continuum can be obtained from the Kubo formula [61, 95] using the nonequilibrium Keldysh Green's function:

$$ \frac{\sigma_1(\omega)}{\sigma_0} = -\frac{1}{8\omega} \int dE \operatorname{Im} \operatorname{Tr} \{ \tau_3 g^R(E) \tau_3 g^K(E - \omega) + \tau_3 g^K(E) \tau_3 g^A(E - \omega) \}, \quad (16) $$

where $g^K = g^RF - Fg^A$, with $F = F_L\tau_0 + F_T\tau_3$, and $\tau_i$ are Pauli matrices in Nambu space. The functions $F_L$ and $F_T$ are the longitudinal (odd in energy) and transverse (even in energy) components of the distribution function, respectively. Both vanish in thermal equilibrium at zero temperature.

In the nonequilibrium case due to quasiparticle generation, the total quasiparticle concentration remains small, so we neglect the back-action of these quasiparticles on $g^{R,A}$, using the equilibrium solution of the Usadel equation. We further assume that the transverse component

$F_T$, associated with particle-hole imbalance [95, 96], relaxes quickly and can be set to zero. Thus, the nonequilibrium effect enters solely through $F_L$, which becomes:

$$ F_L(E) = \tanh(E/2T) - 2f_s(E) \xrightarrow{T \to 0} 1 - 2f_s(E), $$

where $f_s(E)$ is the steady-state quasiparticle distribution function. Substituting this into Eq. (16), one arrives at the simplified expression derived in Eq. (47) of Ref. [61], now applied to a non-equilibrium quasiparticle distribution (with $E_{\pm} = E \pm \omega/2$):

$$ \frac{\sigma_1(\omega)}{\sigma_0} = \frac{1}{\omega} \int_0^{+\infty} dE [F_L(E_+) - F_L(E_-)] [\text{Re} g_{11}^R(E_+) \text{Re} g_{11}^R(E_-) + \text{Im} g_{12}^R(E_+) \text{Im} g_{12}^R(E_-)]. \quad (17) $$

As explained above Eq. (9), an external agent generates quasiparticles that eventually relax into localized states at energies $E \sim E_C$, resulting in a distribution function approximated by $f_s(E) \approx \frac{1}{2}\theta(E_C - |E|)$. Owning to this fact, and to the steep energy dependence of the localized density of states, the integrand in Eq. (17) is sharply peaked near $E = E_C$. To evaluate the integral, we average over fluctuations of the order parame-

ter: $g^R(E_−)$ is approximated using the optimal fluctuation form from Eq. (15), while $g^R(E_+)$ is taken in the uniform (mean-field) approximation given by Eq. (14). This is justified because the dominant fluctuations below the gap that contribute to the conductivity do not significantly modify the continuum states at $E > E_g$. Substituting into Eq. (17) and performing the averaging yields:
$$ \frac{\sigma_1(\omega)}{\sigma_0} = \frac{2}{\omega} \int_0^\infty dE [f_s(E_-) - f_s(E_+)] \frac{\nu_{\text{loc}}(E_-)}{\nu_0} \frac{\nu_{\text{AG}}(E_+)}{\nu_0} \sim \frac{1}{\omega} \int_0^{E_C} dE \frac{\nu_{\text{loc}}(E)}{\nu_0} \frac{\nu_{\text{AG}}(E + \omega)}{\nu_0}, \quad (18) $$

The last term is a smooth function of energy, so the integral can be estimated using Eq. (10), yielding:

$$ \frac{\sigma_1(\omega)}{\sigma_0} \sim \frac{\nu_{\text{AG}}(E_C + \omega)/\nu_0}{\nu_0 \omega r_{\text{loc}}^2(E_C) \min\{d, r_{\text{loc}}(E_C)\}}. \quad (19) $$

The frequency dependence of this result arises from the shape of the smeared BCS peak $\nu_{AG}$, which governs the availability of final states for the ionization process. The corresponding quality factor is then estimated using Eq. (13):

$$ \begin{aligned} Q_{\text{ioniz}}^{\text{dirty}} & \sim \frac{b^2}{\alpha} \frac{\nu_0 \omega}{\nu_{AG} T_c} \frac{\hbar \sigma_2}{e^2} \min\{d, b\xi_0\}, & (20) \\ & \sim (0.05-2) \times 10^{-4} \frac{f[\text{GHz}]}{\alpha T_c [\text{K}]} \sigma_2 [\text{S/m}] & \\ & \times \min\left\{\frac{d}{5}, \xi_0\right\} [\text{nm}]. & (21) \end{aligned} $$

Here, we restore $\hbar$ and use $D/\xi_0^2 = \Delta_0 \sim T_c$, $\sigma_0 = 2\nu_0 e^2 D$, and $r_{\text{loc}} \approx b\xi_0$. For the numerical estimate, Eq. (21), we take $\nu_0$ as the typical value of $\nu_{AG}$ away from the spectral edge [62], and assume $b=3-10$, which accounts for the given range in the prefactor. We use $b=5$ within the minimum function to approximate the thickness dependence.

The exact ratio $\nu_0/\nu_{AG}$ depends not only on $\omega$ and $E_C$, but also on the precise shape of the nonequilibrium distribution $\delta f$, which causes additional smearing of $\nu_{AG}$. Equivalently, one may interpret this as an uncertainty in the definition of $E_C$ that is comparable to the BCS peak width. This leads to an effective $\nu_0/\nu_{AG} \sim 1$.

To contextualize Eq. (21) in terms of Fig. 1 of the main text, we consider TiN devices (see Supplementary Materials Table I or Ref. [97]), which yield $Q_{\text{TiN}}^{\text{dirty}} \sim (0.03-1.2) \times 10^6$ for $\xi_0 \approx 20$ nm. The upper end of this range aligns well with the experimental value for this film. This indicates that the proposed mechanism can account for a significant fraction of the dissipation observed in state-of-the-art dirty superconducting devices.

Nevertheless, the uncertainty in the value of $b$ – arising from the poorly understood kinetics of localized quasiparticles– necessitates further investigation.

### 5. Direct ionization in the clean limit when $\alpha \ll 1$

We expect that Eqs. (19)–(21) remain qualitatively valid in the clean limit, $\xi_0 \ll l$, as well. In this regime, the derivation of $\sigma_1$ should be reformulated in terms of the direction-resolved Green’s function $g(E, n)$, where $n$ is a unit vector on the Fermi surface, and the corresponding

Eilenberger equation for the retarded/advanced components [70]. Averaging over disorder-induced fluctuations in the order parameter $\Delta(r)$, which are responsible for the existence of localized states, is expected to yield a result similar to Eq. (18), with all subsequent approximations and transformations remaining applicable.

For comparison with experimental data in Fig. 1 of the main text, we must also account for the additional dependence of $Q$ on $\omega$ and $\sigma_2$ via the kinetic inductance fraction $\alpha \ll 1$. For a cubic 3D cavity resonator with side length $L = \pi c/\omega$ corresponding to the fundamental mode, the kinetic inductance fraction can be estimated as:

$$ \frac{1}{\alpha} = \frac{L}{6\lambda} \approx 0.18 \sqrt{\frac{\sigma_2[\text{S/m}]}{f[\text{GHz}]}}. \quad (22) $$

In the clean limit, the film thickness satisfies $d \gg \xi_0$, so Eq. (21) reduces to:

$$ Q_{\text{ioniz}}^{\text{clean}} \sim 2 \times 10^{-5} \sigma_2^{3/2} [\text{S/m}] \cdot \xi_0 [\text{nm}]. \quad (23) $$

In this estimate, we have dropped explicit dependence on $\omega$ and $T_c$, as their influence is expected to be smaller than the overall uncertainty introduced by the chain of approximations. Consequently, it is difficult to assign a reliable error range to Eq. (23).

Nonetheless, this formula predicts quality factors that are at least two orders of magnitude higher than those measured in aluminum cavities with $\sigma_2 = 4 \times 10^9$ S/m and $\xi_0 = 100$ nm, and the discrepancy increases with $\sigma_2$, since the dependence is faster than linear. While the numerical disagreement at specific values of $\sigma_2$ may be attributed to the approximate nature of Eq. (23), the broader trend suggests that dissipation in these systems is dominated by a stronger mechanism –such as delocalized quasiparticles, discussed in Subsec. B 7 below.

### 6. Coherent TLS Rabi oscillations in the clean limit

For low frequencies in clean systems (see Supplementary Materials Figs. 2d and 3), the analysis of Subsec. B 4 must be modified. Instead of using the Kubo formula for direct ionization, Eq. (16), one must calculate $\sigma_1$ arising from an ensemble of coherent two-level systems (TLSs), formed by the first two bound quasiparticle states in a typical local minimum of the order parameter [70].

The power absorbed by a single TLS with energy splitting $\omega_0$, dephasing rate $\Gamma_2$, and relaxation rate $\Gamma_1$, is
described by a Lorentzian profile [98, Sec. 2.3]:

$$P = \frac{1}{2} (\mathbf{p} \cdot \mathbf{E})^2 \omega \tanh \left(\frac{\omega_0}{2T}\right) \frac{\Gamma_2}{(\omega_0 - \omega)^2 + \Gamma_2^2}, \quad (24)$$

where $\mathbf{p}$ is the transition dipole moment and $\mathbf{E}$ is the applied electric field. At zero temperature ($T=0$), an ensemble of such TLSs with randomly oriented dipole moments contributes to the dissipative conductivity as

$$\sigma_1^{\text{TLS}}(\omega) = \int d\omega_0 n(\omega_0) \frac{\bar{p}^2(\omega_0)}{3} \frac{\omega\Gamma_2/2}{(\omega_0 - \omega)^2 + \Gamma_2^2}, \quad (25)$$

where $\bar{p}(\omega_0)$ is the typical dipole moment of a TLS with splitting $\omega_0$, and $n(\omega_0)$ is the density of such TLSs. The latter is determined by the number of localized quasiparticles occupying the ground state of local dips with excitation energy $\omega_0 \approx \zeta\epsilon$, and is given by:

$$n(\omega_0) = \frac{1}{\zeta} f_s(E) \nu_{\text{loc}}(E)|_{\epsilon \equiv E_g - E = \omega_0 / \zeta}, \quad (26)$$

where $\zeta \approx 0.7$ [70] is the ratio between the first excitation energy and the ground state energy in a typical optimal well. As before, we assume the steady-state distribution $f_s(E) \approx \theta(E_C - E)/2$, cf. Eq. (9).

The transition dipole moment for such a TLS is estimated as

$$\bar{p}(\omega_0) \sim e \frac{\varepsilon}{\Delta_0} r_{\text{loc}}(E) \Big|_{\varepsilon = \omega_0 / \zeta}, \quad (27)$$

where the prefactor $\varepsilon/\Delta_0$ accounts for the near equal mixing of electron and hole components for quasiparticles near the gap edge.

Since $\Gamma_2 \ll \varepsilon_T$ (the typical energy scale of the localized DOS [70]), the Lorentzian is sharply peaked and the integral simplifies:

$$\sigma_1^{\text{TLS}}(\omega) \sim \frac{\pi\omega}{6\zeta} f_s(E) \nu_{\text{loc}}(E) \left[ e \frac{\varepsilon}{\Delta_0} r_{\text{loc}}(E) \right]^2, \quad (28)$$

with $\varepsilon \equiv E_g - E = \omega/\zeta$.

We emphasize that this estimate is considerably less reliable than Eq. (19) for the dirty limit. The frequency dependence is sensitive to the detailed shape of both $f_s(E)$ and $\nu_{\text{loc}}(E)$, and anisotropic features of the optimal fluctuations in the clean limit [70] are not accounted for –neither in the kinetic treatment nor in the transition matrix elements.

We now employ the specific profile (4) of $\nu_{\text{loc}}$ in Eq. (10) to express the integral in terms of $\nu_{\text{loc}}(E_C)$:

$$\varepsilon_T V_{\text{loc}}(E_C) \nu_{\text{loc}}(E_C) \frac{4}{5} \left( \frac{\varepsilon_C}{\varepsilon_T} \right)^{-1/4} \sim 1, \quad (29)$$

which allows us to estimate the quality factor at $\omega \sim$

$\zeta(E_g - E_C)$, where $\sigma_1$ is maximal:

$$Q_{\text{TLS}}^{\text{clean}} \sim (3 - 10) \frac{1}{\alpha} \left( \frac{\Delta_0}{\varepsilon_C} \right)^2 \frac{\hbar \sigma_2 \xi_0}{e^2}. \quad (30)$$

Here, we use that $\varepsilon_C = E_g - E_C \sim \varepsilon_T$ within our precision, and take $b = 3 - 10$ as before. We also assume the condition $d \gg b\xi_0$ holds, which is typical for 3D resonators. However, sufficiently thin Al films may violate this assumption (see Supplementary Materials Table I).

In analogy to Eq. (23), the dependence of $\alpha$ in the clean limit [cf. Eq. (22)] should be included, leading to:

$$Q_{\text{TLS}}^{\text{clean}} \sim (0.2 - 3) \times 10^{-3} \frac{\sigma_2^{3/2} [\text{S/m}] \xi_0 [\text{nm}]}{\sqrt{f [\text{GHz}]}}. \quad (31)$$

In this estimate, we used $\varepsilon_C/\Delta_0 = 0.05-0.1$ as the characteristic depth of the relevant order parameter fluctuations. Both this uncertainty and that in $b$ contribute to an overall uncertainty of about an order of magnitude.

It is instructive to compare this estimate to the contribution from direct ionization:

$$\frac{Q_{\text{ioniz}}^{\text{clean}}}{Q_{\text{TLS}}^{\text{clean}}} \sim b^2 \left( \frac{\varepsilon_C}{\Delta_0} \right)^2 \frac{\omega}{T_c} \ll 1. \quad (32)$$

Although the quantitative accuracy of this estimate is limited due to the approximate nature of both individual expressions, the ratio is expected to be well below unity –even in the worst-case scenario with $b \sim 10$, $\varepsilon_C/\Delta_0 \sim 0.1$, and $\omega \sim 0.1 T_c$. This supports the general picture presented in Supplementary Materials Fig. 3, though we cannot exclude the possibility that both contributions may be comparable in certain conditions.

The presence of TLS-like subsystems of various microscopic origins in superconducting devices has received considerable attention in the literature [11, 70], prompting the development of advanced experimental techniques to probe these degrees of freedom, alongside significant theoretical efforts to describe the emergent phenomenology. In particular, the standard framework for analyzing the power dependence of the quality factor [11, Eq. (22)] is expected to be applicable in this context.

## 7. Dissipation due to delocalized quasiparticles

Having discussed the main possibilities involving localized quasiparticles, we now turn to a more straightforward scenario: either due to a high effective quasiparticle temperature $T_{\text{eff}}$ or a sufficiently strong external source, quasiparticles may occupy delocalized states in the bulk of the excitation spectrum in the steady state (see sub-sec. B 2).

In the dirty limit, such quasiparticles contribute to the dissipative conductivity via an expression analogous to Eq. (18), with $\nu_{\text{loc}}(E)$ replaced by $\nu_{\text{AG}}(E)$, the appropriate density of initial states. Neglecting the frequency de-
pendence of the final density of states immediately leads to the Mattis-Bardeen-type result (12). A more careful evaluation in the case of a quasi-equilibrium distribution function, Eq. (8), yields a numerical prefactor for Eq. (12) that depends on $\eta$, $T_{\text{eff}}/\Delta_0$, and $\omega/\Delta_0$ [61, see Eqs. (58)–(63)]. The estimate (12) is, in fact, also applicable in the clean limit [35, 61].

For the dirty limit and the case when $\omega$ is large enough that the final state lies beyond the BCS coherence peak (regardless of $T_{\text{eff}}$), the quality factor is estimated using Eqs. (59) and (62) of Ref. [61] for $\sigma_1$:

$$Q_{\text{MT}}^{\text{dirty}} \approx \frac{1}{\alpha} \frac{1}{\sqrt{2}} \left(\frac{\omega}{\Delta}\right)^{3/2} \frac{1}{n_{\text{qp}}\xi_0^3} \frac{\hbar\sigma_2\xi_0}{e^2}, \quad (33)$$

where we have used $\sigma_0 = 2\nu_0e^2D$ and assumed the 3D limit $d \gg \xi_0$ for simplicity.

Crucially, the value of $n_{\text{qp}}\xi_0^3$ for delocalized quasiparticles is difficult to predict theoretically, as the recombination rate balancing the external quasiparticle generation exhibits a much smoother energy dependence and is highly sensitive to the specific microscopic details.

One can, however, use Eq. (33) to estimate the concentration of delocalized quasiparticles required to account for the observed dissipation. For the best Al cavities in Fig. 1, this yields $n_{\text{qp}}\xi_0^3 \sim 0.3$. This value is roughly two orders of magnitude larger than the estimate in Eq. (11) for localized quasiparticles, echoing the result obtained for localized quasiparticles in the clean limit, Eq. (23), despite the limited accuracy of that estimate. One thus concludes that while modern devices have made significant progress in suppressing excess quasiparticles, their localization is not yet achieved, even for the best-performing cavities represented on the right side of Fig. 1.

We also note that the evolution of the distribution function of delocalized quasiparticles with applied power has recently been described comprehensively from a theoretical standpoint [99], paving the way for a more accurate understanding of the observed power dependence of the quality factor. In particular, Ref. [99] demonstrates that the specific form of this dependence is sensitive to both the properties of the quasiparticle-generating agent and material parameters, making a simple universal description unlikely.

8. Other possible loss mechanisms due to nonequilibrium quasiparticles at low frequencies

While most modern devices arguably operate in the regime $\omega > \varepsilon_T$, where dissipation is of the Mattis-Bardeen type [Eq. (12)], certain cases feature frequencies low enough to render alternative mechanisms relevant [70], with quasiparticle TLSs discussed above being one example. It is also possible that, in the dirty limit, the number of bound states in a typical local dip of the order parameter exceeds one [100]. However, as pointed out in Ref. [70], direct transitions between these bound

states are expected to be inefficient in the dirty case due to their small transition dipole moments.

This nevertheless leaves room for several other dissipation mechanisms, illustrated in Supplementary Materials Figs. 2e and 3. If the actual level spacing between bound states within a well is small enough, the local spectrum may effectively behave as continuous, resulting in Mattis-Bardeen-like dissipation within each individual well. Alternatively, slow relaxation rates of localized quasiparticles (see subsec. B 2) may favor nonequilibrium dynamics within a single well, resembling the mechanism behind the enhanced dissipative response in thin superconducting films in the mixed state [93]. Due to the altered quasiparticle distribution function in such a scenario, the power dependence of the quality factor is expected to become even more pronounced.

Finally, another intriguing possibility is photon-assisted variable-range hopping between different local dips of the order parameter (see Supplementary Materials Fig. 2f), a mechanism that, to our knowledge, has not been systematically studied. While the problem bears some resemblance to its semiconductor analog [101], the strong energy dependence of the density of localized states, Eqs. (4)–(5), must be taken into account.

C. Dissipation by an isolated Vortex

The inverse quality factor $1/Q$ of a resonator in the presence of a low magnetic field is determined by contributions from individual vortices that penetrate the superconducting film and become pinned by disorder. In this low-field regime, variations in the effective superfluid stiffness $\Theta(B)$ are negligible. Thus, the dissipation contribution to $\frac{1}{Q} = \frac{\sigma_1}{\sigma_2}$ is governed by the behavior of the real part of the conductivity $\sigma_1(\omega)$. The complex sheet conductivity of a superconductor in the mixed state with pinned vortices is given by (see e.g., Ref. [93, 102]):

$$\sigma^{\square}(\omega) = \frac{id}{\omega\mu_0\lambda_L^2 + \frac{1}{\mu_0(k_0 - i\omega\eta)}}. \quad (34)$$

In the above expression, $\lambda_L$ is the London penetration depth, $d$ is the film thickness, $k_0$ is the curvature of the vortex pinning potential, and $\eta_v$ is the vortex friction coefficient, determining the friction force $f_v = \eta_v v$ acting on a vortex moving at velocity $v$. The friction coefficient $\eta_v$ is related to the bulk DC conductivity in the flux-flow regime (at currents well above the depinning threshold) via $\sigma_{ff} = \frac{\eta_v}{B\Phi_0d}$. It is further connected to the normal-state conductivity $\sigma_n$ by

$$\eta_v = \zeta\Phi_0H_{c2}\mu_0\sigma_n d,$$

where $\zeta \sim 1$ is a numerical factor.

Expanding the real part of $\sigma$ from Eq. (34) in the small
B limit, we find:

$$ \sigma_1^\square = \frac{\Phi_0 B \eta_v d^2}{\mu_0^2 \lambda_L^4 k_0^2 \zeta} \sim \frac{\sigma_n B \Theta}{\Phi_0 j_c^2 \lambda_L^2}, \quad (35) $$

where $j_c$ is the critical current density of the material, $\Theta = \Phi_0^2 d/(4\pi^2\mu_0\lambda_L^2)$ is the superfluid stiffness, and we have used the approximate relations $\Phi_0 j_c d \approx k_0 \xi_0$ and $H_{c2} \approx \Phi_0/2\pi\mu_0\xi_0^2$. Using Eq. (35) together with the standard expression for the pair-breaking current density $j_0 = \Phi_0/(3\sqrt{3}\pi\mu_0\lambda_L^2\xi_0)$, we can express the vortex-induced contribution to the inverse quality factor as:

$$ \frac{1}{Q_{\text{Vortex}}} \sim 10 \frac{\sigma_n d}{\sigma_2^\square \mu_0 H_{c2}} \left(\frac{j_0}{j_c}\right)^2. \quad (36) $$

Considering a residual field strength $B_0 \sim 10^{-7}$ T, the factor $B_0/\mu_0 H_{c2} \sim 10^{-6} - 10^{-8}$ for different materials. The last factor in Eq. (36) is very much material-dependent and can reach $10^4 - 10^6$. This non-universality of the prefactor of $\sigma_2$ in Eq. (36) is also reflected in the fact that highly disordered films exhibit strong vortex pinning, which limits vortex-induced dissipation, in contrast to low-disorder films, where weaker pinning results in a higher $j_0/j_c$ ratio. According to Eq. (36) one could expect rather low Q-factor due to vortices in low-disorder films; however, i) at low T vortices are energetically unfavorable at very low external fields $B \ll \Phi_0/w^2$ where $w \sim$ few microns is typical width of a resonator stripe [103], and ii) precisely due to weak pinning in low-disordered material, vortices are not pinned in such a narrow stripes. Thus they are expelled out of a stripe to decrease energy.
<figure><img src="image_3.png" /><figcaption>Supplementary Materials Fig. 1. **Frequency histogram of the datasets.** Histogram of the frequencies corresponding to the datasets shown in Fig. 1.</figcaption></figure>
<figure><img src="image_4.png" /></figure>

Supplementary Materials Fig. 2. Dissipation mechanisms induced by nonequilibrium quasiparticles. a, Sketch of the quasiparticle density of states $\nu(E)$ as a function of energy near the superconducting gap edge. $\nu_0$ denotes the normal-state density of states per spin projection at the Fermi level; $\Delta_0$ is the superconducting gap in the clean limit; and $E_g \approx \Delta_0(1-\eta^{2/3})^{3/2}$ marks the mobility edge separating the delocalized states (pink) from the localized tail. The difference between $E_g$ and $\Delta_0$ arises from disorder-induced smearing of the BCS coherence peak, characterized by the depairing parameter $\eta$ [62, 74]. $\nu_{\text{loc}}(E)$ represents the subgap tail of localized quasiparticle states, with a characteristic energy scale $\epsilon_T \ll \Delta_0$, and $E_C$ denotes the energy threshold below which quasiparticle recombination becomes inefficient [69]. b–f, Schematic representation of possible dissipation mechanisms involving nonequilibrium quasiparticles, approximately ordered by decreasing dissipation strength (see also Fig. 3): b, Standard Mattis-Bardeen-like transitions in the continuum of delocalized states [35, 61], subsec. B7; c, Ionization of a bound quasiparticle state in a local well of suppressed superconducting order parameter (blue) with characteristic size $r_{\text{loc}}(E_C)$, subsecs. B4, B5; d, Coherent Rabi oscillations between two bound states within a single well in the clean limit, subsec. B6; e,f, Semiclassical dynamics involving multiphoton absorption and photon-assisted variable-range hopping between different localized wells, potentially in nonequilibrium regimes, subsec. B8.

<figure><img src="image_5.png" /></figure>

Supplementary Materials Fig. 3. Classification of the main dissipation mechanisms due to nonequilibrium quasiparticles. Green shading indicates the dominant dissipation mechanism in dirty superconducting thin films (left side of Fig. 1, main text), as discussed in subsec. B4. Orange indicates the dominant mechanism in clean films and 3D cavities (right side of Fig. 1, main text; see subsec. B7). A red star marks those mechanisms for which a power-dependence analysis of the quality factor is available.
Supplementary Materials Table I: Table of resonator and qubit parameters for the data shown in Fig. 1 and Supplementary Materials Fig. 1. The resonators have either planar geometry (thickness *d*) or are 3D cavities. Quality factors are measured at low temperatures (typically *T* ~ 20 mK) and in the low-photon-number regime, at frequency *f*. Material properties such as critical temperature *T*<sub>c</sub> and sheet normal-state resistance *R*<sub>□</sub> (measured above the superconducting transition) depend on the material, its purity (residual-resistivity ratio, RRR), and the film thickness. Resistance values reported only at room temperature are labeled (RT). In the dirty limit, the sheet kinetic inductance can be calculated using *L*<sub>K</sub> = *ħ*R<sub>□</sub>/(πΔ), where Δ = 1.76 *kT*<sub>c</sub> for conventional BCS superconductors. *L*<sub>K</sub> can also be inferred from electromagnetic simulations, from fits to the Mattis-Bardeen theory [35] based on the temperature dependence of the resonance frequency *f*(*T*), or from fits to the plasmon dispersion relation [27]. In the clean limit, we use tabulated values for the London penetration depth λ for bulk superconductors, with thickness corrections applied when available. Data that are partially missing but can be reasonably estimated from references are marked with a star (*). The estimation of λ and *L*<sub>K</sub> for cleaner superconductors (e.g., Nb, Al, Ta) is often complicated by the absence of transport characterization and is therefore subject to larger uncertainties. The *Q*<sub>i</sub> value of the grAl transmon from Ref. [78] is obtained as the mean of the different measurement runs for the same device reported in Table II of that work.

<table><thead><tr><th>Material</th><th>Ref</th><th>d (nm)</th><th>R<sub>□</sub> (Ω/□)</th><th>T<sub>c</sub> (K)</th><th>f (GHz)</th><th>λ (nm)</th><th>L<sub>K</sub> (pH/□)</th><th>σ<sub>2</sub> (×10<sup>8</sup> S/m)</th><th>α</th><th>Q<sub>i</sub> (×10<sup>5</sup>)</th></tr></thead><tbody><tr><td>Al</td><td>[72]</td><td>60</td><td>0.15</td><td>1.17</td><td>5.289</td><td>92</td><td></td><td>28</td><td></td><td>25</td></tr><tr><td>Al</td><td>[104]</td><td>150</td><td></td><td></td><td>5.244</td><td>65</td><td>[105]</td><td>57</td><td></td><td>7.9</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>3.83</td><td>80</td><td>[105]</td><td>51.6</td><td>8.210<sup>-2</sup></td><td>1.6</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>6.129</td><td>80</td><td>[105]</td><td>32</td><td>1.7810<sup>-2</sup></td><td>3</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>3.81</td><td>80</td><td>[105]</td><td>51.9</td><td>8.210<sup>-2</sup></td><td>6.6</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>6.08</td><td>80</td><td>[105]</td><td>32.5</td><td>1.7810<sup>-2</sup></td><td>7.2</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>4.97</td><td>80</td><td>[105]</td><td>39.8</td><td>4.310<sup>-2</sup></td><td>5.3</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>6.12</td><td>80</td><td>[105]</td><td>32.3</td><td>1.7810<sup>-2</sup></td><td>7.6</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>3.77</td><td>80</td><td>[105]</td><td>52.5</td><td>8.210<sup>-2</sup></td><td>7.5</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>6.125</td><td>80</td><td>[105]</td><td>32.3</td><td>1.7810<sup>-2</sup></td><td>8</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>3.87</td><td>80</td><td>[105]</td><td>51.1</td><td>8.210<sup>-2</sup></td><td>11.5</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>6.12</td><td>80</td><td>[105]</td><td>32.3</td><td>1.7810<sup>-2</sup></td><td>9.2</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>3.76</td><td>80</td><td>[105]</td><td>52.6</td><td>8.210<sup>-2</sup></td><td>11</td></tr><tr><td>Al</td><td>[31]</td><td>100</td><td></td><td></td><td>6.12</td><td>80</td><td>[105]</td><td>32.3</td><td>1.7810<sup>-2</sup></td><td>17.2</td></tr><tr><td>Al</td><td>[106]</td><td>100</td><td></td><td></td><td>4.487</td><td>80</td><td>[105]</td><td>44</td><td>1.810<sup>-2</sup></td><td>3.47</td></tr><tr><td>Al</td><td>[106]</td><td>100</td><td></td><td></td><td>4.501</td><td>80</td><td>[105]</td><td>44</td><td>1.810<sup>-2</sup></td><td>7.21</td></tr><tr><td>Al</td><td>[106]</td><td>100</td><td></td><td></td><td>4.512</td><td>80</td><td>[105]</td><td>44</td><td>1.810<sup>-2</sup></td><td>7.17</td></tr><tr><td>Al</td><td>[106]</td><td>100</td><td></td><td></td><td>4.506</td><td>80</td><td>[105]</td><td>44</td><td>1.810<sup>-2</sup></td><td>9.85</td></tr><tr><td>Al</td><td>[107]</td><td>100</td><td></td><td></td><td>5.606</td><td>80</td><td>[105]</td><td>35</td><td></td><td>7.5</td></tr><tr><td>Al</td><td>[38]</td><td>120</td><td></td><td></td><td>6.7</td><td>80</td><td>[105]</td><td>29.5</td><td>5.510<sup>-2</sup></td><td>0.6</td></tr><tr><td>Al</td><td>[38]</td><td>120</td><td></td><td></td><td>7</td><td>80</td><td>[105]</td><td>28.2</td><td>5.510<sup>-2</sup></td><td>0.6</td></tr><tr><td>Al</td><td>[38]</td><td>120</td><td></td><td></td><td>7.1</td><td>80</td><td>[105]</td><td>27.8</td><td>2.810<sup>-2</sup></td><td>11</td></tr><tr><td>Al</td><td>[38]</td><td>120</td><td></td><td></td><td>7</td><td>80</td><td>[105]</td><td>28.2</td><td>5.510<sup>-2</sup></td><td>0.7</td></tr><tr><td>Al</td><td>[38]</td><td>120</td><td></td><td></td><td>7</td><td>80</td><td>[105]</td><td>28.2</td><td>3.510<sup>-2</sup></td><td>11</td></tr><tr><td>Al</td><td>[38]</td><td>120</td><td></td><td></td><td>7.1</td><td>80</td><td>[105]</td><td>27.8</td><td>1.810<sup>-2</sup></td><td>17</td></tr><tr><td>Al</td><td>[108]</td><td>500</td><td></td><td></td><td>Fig.2</td><td>52</td><td>[105]</td><td></td><td>8.110<sup>-3</sup></td><td>Fig.2</td></tr><tr><td>Al</td><td>[108]</td><td>300</td><td></td><td></td><td>Fig.2</td><td>52</td><td>[105]</td><td></td><td>8.110<sup>-3</sup></td><td>Fig.2</td></tr><tr><td>Al</td><td>[108]</td><td>150</td><td></td><td></td><td>Fig.2</td><td>65</td><td>[105]</td><td></td><td>1.010<sup>-2</sup></td><td>Fig.2</td></tr><tr><td>Al Transmon</td><td>[108]</td><td>500</td><td></td><td></td><td>Tab. II</td><td>52</td><td>[105]</td><td></td><td></td><td>Tab. II</td></tr><tr><td>Al Transmon</td><td>[108]</td><td>300</td><td></td><td></td><td>Tab. II</td><td>52</td><td>[105]</td><td></td><td></td><td>Tab. II</td></tr><tr><td>Al Transmon</td><td>[108]</td><td>150</td><td></td><td></td><td>Tab. II</td><td>65</td><td>[105]</td><td></td><td></td><td>Tab. II</td></tr><tr><td>Al Transmon</td><td>[81]</td><td>120</td><td></td><td></td><td>Tab. I</td><td>63</td><td>[105]</td><td></td><td></td><td>Tab. I</td></tr><tr><td>Al Transmon</td><td>[109]</td><td>200</td><td></td><td></td><td>5*</td><td></td><td>~ 100*</td><td></td><td></td><td>Fig. 2b</td></tr><tr><td>Al susp. strips</td><td>[110]</td><td>2000</td><td></td><td></td><td>5*</td><td>52</td><td></td><td>93.7*</td><td>310<sup>-5</sup></td><td>440</td></tr><tr><td>Al susp. strips</td><td>[110]</td><td>2000</td><td></td><td></td><td>5*</td><td>52</td><td></td><td>93.7*</td><td>310<sup>-5</sup></td><td>480</td></tr><tr><td>Al susp. strips</td><td>[110]</td><td>600</td><td></td><td></td><td>5*</td><td>52</td><td></td><td>93.7*</td><td>310<sup>-5</sup></td><td>370</td></tr></tbody></table>

*Continued on next page*
<table><caption>Supplementary Materials Table I – Continued from previous page</caption><thead><tr><th>Material</th><th>Ref</th><th>d (nm)</th><th>R<sub>□</sub> (Ω/□)</th><th>T<sub>c</sub> (K)</th><th>f (GHz)</th><th>λ (nm)</th><th>L<sub>K</sub> (pH/□)</th><th>σ<sub>2</sub> (×10<sup>8</sup> S/m)</th><th>α</th><th>Q<sub>i</sub> (×10<sup>5</sup>)</th></tr></thead><tbody><tr><td>Al susp. strips</td><td>[110]</td><td>800</td><td></td><td></td><td>5*</td><td>52</td><td></td><td>93.7*</td><td>3 10<sup>-5</sup></td><td>370</td></tr><tr><td>Al susp. strips</td><td>[110]</td><td>600</td><td></td><td></td><td>5*</td><td>52</td><td></td><td>93.7*</td><td>3 10<sup>-5</sup></td><td>1080</td></tr><tr><td>Al susp. strips</td><td>[110]</td><td>600</td><td></td><td></td><td>5*</td><td>52</td><td></td><td>93.7*</td><td>3 10<sup>-5</sup></td><td>1270</td></tr><tr><td>Al susp. strips</td><td>[110]</td><td>600</td><td></td><td></td><td>5*</td><td>52</td><td></td><td>93.7*</td><td>3 10<sup>-5</sup></td><td>96</td></tr><tr><td>Al susp. strips</td><td>[110]</td><td>800</td><td></td><td></td><td>5*</td><td>52</td><td></td><td>93.7*</td><td>3 10<sup>-5</sup></td><td>370</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>9.613</td><td>65</td><td></td><td>31.5</td><td></td><td>51</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>9.464</td><td>65</td><td></td><td>31.6</td><td></td><td>56</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>9.478</td><td>65</td><td></td><td>31.6</td><td></td><td>48</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>7.69</td><td>65</td><td></td><td>39</td><td></td><td>24</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>11.448</td><td>65</td><td></td><td>26</td><td></td><td>10</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>11.416</td><td>65</td><td></td><td>26</td><td></td><td>1.5 10<sup>3</sup></td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>11.45</td><td>65</td><td></td><td>26</td><td></td><td>150</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>9.455</td><td>52</td><td></td><td>49.5</td><td></td><td>420</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>9.481</td><td>52</td><td></td><td>49.4</td><td></td><td>430</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>9.481</td><td>52</td><td></td><td>49.4</td><td></td><td>690</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>7.7</td><td>52</td><td></td><td>60.8</td><td></td><td>310</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>11.448</td><td>52</td><td></td><td>40.9</td><td></td><td>140</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>11.417</td><td>52</td><td></td><td>41.0</td><td></td><td>6.09 10<sup>3</sup></td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>11.419</td><td>52</td><td></td><td>41.0</td><td></td><td>320</td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>11.44</td><td>52</td><td></td><td>40.9</td><td></td><td>7.4 10<sup>3</sup></td></tr><tr><td>Al (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>11.442</td><td>52</td><td></td><td>40.9</td><td></td><td>5.2 10<sup>3</sup></td></tr><tr><td>Al alloy (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>9.513</td><td>194</td><td></td><td>3.53</td><td></td><td>51</td></tr><tr><td>Al alloy (3D)</td><td>[111]</td><td>bulk</td><td></td><td>1.18</td><td>9.450</td><td>194</td><td></td><td>3.56</td><td></td><td>32</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>7.43</td><td>62</td><td></td><td>44.3</td><td>5 10<sup>-5</sup></td><td>830</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>7.417</td><td>62</td><td></td><td>44.4</td><td>5 10<sup>-5</sup></td><td>660</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>7.425</td><td>62</td><td></td><td>44.3</td><td>5 10<sup>-5</sup></td><td>810</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>7.417</td><td>62</td><td></td><td>44.4</td><td>5 10<sup>-5</sup></td><td>930</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>7.427</td><td>62</td><td></td><td>44.3</td><td>5 10<sup>-5</sup></td><td>790</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>7.428</td><td>62</td><td></td><td>44.3</td><td>5 10<sup>-5</sup></td><td>910</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>7.427</td><td>62</td><td></td><td>44.3</td><td>5 10<sup>-5</sup></td><td>820</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>6.476</td><td>62</td><td></td><td>50.8</td><td>5 10<sup>-5</sup></td><td>860</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>5.478</td><td>62</td><td></td><td>60.1</td><td>5 10<sup>-5</sup></td><td>940</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>4.5</td><td>62</td><td></td><td>73.2</td><td>5 10<sup>-5</sup></td><td>1.15 10<sup>3</sup></td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>5.932</td><td>62</td><td></td><td>55.5</td><td>5 10<sup>-5</sup></td><td>300</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>7.437</td><td>62</td><td></td><td>44.3</td><td>5 10<sup>-5</sup></td><td>1.01 10<sup>3</sup></td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>5.928</td><td>62</td><td></td><td>55.5</td><td>5 10<sup>-5</sup></td><td>750</td></tr><tr><td>Al (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>5.923</td><td>62</td><td></td><td>55.6</td><td>5 10<sup>-5</sup></td><td>840</td></tr><tr><td>Al alloy (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>5.976</td><td>189</td><td></td><td>5.933</td><td></td><td>50</td></tr><tr><td>Al alloy (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>5.929</td><td>189</td><td></td><td>5.98</td><td></td><td>120</td></tr><tr><td>Al alloy (3D)</td><td>[112]</td><td>bulk</td><td></td><td>1.18</td><td>5.939</td><td>189</td><td></td><td>5.97</td><td></td><td>70</td></tr><tr><td>Al (3D)</td><td>[113]</td><td>bulk</td><td></td><td></td><td>6.389</td><td>235</td><td></td><td>3.58</td><td>4.61 10<sup>-5</sup></td><td>250</td></tr><tr><td>Al (3D)</td><td>[113]</td><td>bulk</td><td></td><td></td><td>6.75</td><td>37.6</td><td></td><td>132</td><td>1.89 10<sup>-5</sup></td><td>100</td></tr><tr><td>Al alloy (3D)</td><td>[114]</td><td>bulk</td><td></td><td>1.18</td><td>2.5123</td><td>194</td><td></td><td>13.39</td><td></td><td>260</td></tr><tr><td>Al alloy (3D)</td><td>[114]</td><td>bulk</td><td></td><td>1.18</td><td>2.8156</td><td>194</td><td></td><td>11.95</td><td></td><td>220</td></tr><tr><td>Al alloy (3D)</td><td>[114]</td><td>bulk</td><td></td><td>1.18</td><td>3.5571</td><td>194</td><td></td><td>9.46</td><td></td><td>236</td></tr><tr><td>Al alloy (3D)</td><td>[114]</td><td>bulk</td><td></td><td>1.18</td><td>4.0007</td><td>194</td><td></td><td>8.41</td><td></td><td>247</td></tr><tr><td>Al alloy (3D)</td><td>[114]</td><td>bulk</td><td></td><td>1.18</td><td>4.2245</td><td>194</td><td></td><td>7.96</td><td></td><td>254</td></tr><tr><td>Al alloy (3D)</td><td>[114]</td><td>bulk</td><td></td><td>1.18</td><td>4.2248</td><td>194</td><td></td><td>7.96</td><td></td><td>88</td></tr><tr><td>grAl</td><td>[115]</td><td>20</td><td>1153</td><td>1.6</td><td>4.04</td><td></td><td>995</td><td>0.019</td><td>0.99</td><td>1.26</td></tr><tr><td>grAl</td><td>[115]</td><td>20</td><td>382</td><td>1.6</td><td>3.11</td><td></td><td>329</td><td>0.077</td><td>0.98</td><td>0.37</td></tr><tr><td>grAl</td><td>[116]</td><td>20</td><td></td><td>1.7</td><td>4.05</td><td></td><td>1200</td><td>0.019</td><td>1</td><td>0.2</td></tr><tr><td>grAl</td><td>[116]</td><td>20</td><td></td><td>1.7</td><td>4.05</td><td></td><td>1200</td><td>0.021</td><td>1</td><td>0.13</td></tr><tr><td>grAl</td><td>[116]</td><td>20</td><td></td><td>1.75</td><td>3.88</td><td></td><td>2000</td><td>0.012</td><td>1</td><td>0.09</td></tr><tr><td>grAl</td><td>[117]</td><td>120</td><td>17</td><td>1.8</td><td>3.78</td><td></td><td>20</td><td>0.17</td><td>0.56</td><td>1.0</td></tr><tr><td>grAl</td><td>[118]</td><td>20</td><td></td><td></td><td>5.0</td><td></td><td>700 [119]</td><td>0.022</td><td></td><td>0.773</td></tr><tr><td>grAl</td><td>[118]</td><td>20</td><td></td><td></td><td>5.644</td><td></td><td>700 [119]</td><td>0.020</td><td>0.9</td><td>1.14</td></tr><tr><td>grAl</td><td>[118]</td><td>20</td><td></td><td></td><td>6.247</td><td></td><td>700 [119]</td><td>0.018</td><td>0.9</td><td>1.34</td></tr><tr><td>grAl</td><td>[118]</td><td>20</td><td></td><td></td><td>6.848</td><td></td><td>700 [119]</td><td>0.016</td><td>0.9</td><td>1.59</td></tr></tbody></table>

Continued on next page
<table><caption>Supplementary Materials Table I – Continued from previous page</caption><thead><tr><th>Material</th><th>Ref</th><th>d (nm)</th><th>R<sub>□</sub> (Ω/□)</th><th>T<sub>c</sub> (K)</th><th>f (GHz)</th><th>λ (nm)</th><th>L<sub>K</sub> (pH/□)</th><th>σ<sub>2</sub> (×10<sup>8</sup> S/m)</th><th>α</th><th>Q<sub>i</sub> (×10<sup>5</sup>)</th></tr></thead><tbody><tr><td>grAl</td><td>[118]</td><td>20</td><td></td><td></td><td>7.469</td><td></td><td>700 [119]</td><td>0.015</td><td>0.9</td><td>1.70</td></tr><tr><td>grAl</td><td>[118]</td><td>20</td><td></td><td></td><td>8.273</td><td></td><td>700 [119]</td><td>0.013</td><td>0.9</td><td>0.88</td></tr><tr><td>grAl</td><td>[118]</td><td>20</td><td></td><td></td><td>9.474</td><td></td><td>700 [119]</td><td>0.012</td><td>0.9</td><td>1.25</td></tr><tr><td>grAl</td><td>[118]</td><td>20</td><td></td><td></td><td>10.538</td><td></td><td>700 [119]</td><td>0.010</td><td>0.9</td><td>1.16</td></tr><tr><td>grAl</td><td>[118]</td><td>20</td><td></td><td></td><td>11.310</td><td></td><td>700 [119]</td><td>0.010</td><td>0.9</td><td>1.86</td></tr><tr><td>grAl</td><td>[120]</td><td>50</td><td></td><td></td><td>[120] Tab. I</td><td></td><td>150</td><td></td><td></td><td>[120] Tab. I</td></tr><tr><td>grAl</td><td>[121]</td><td>20</td><td></td><td></td><td>5.78</td><td></td><td>670</td><td>0.020</td><td></td><td>1.19</td></tr><tr><td>grAl</td><td>[121]</td><td>20</td><td></td><td></td><td>4.68</td><td></td><td>670</td><td>0.025</td><td></td><td>2.58</td></tr><tr><td>grAl</td><td>[121]</td><td>20</td><td></td><td></td><td>7.39</td><td></td><td>530</td><td>0.020</td><td></td><td>3.25</td></tr><tr><td>grAl</td><td>[121]</td><td>30</td><td></td><td></td><td>7.02</td><td></td><td>260</td><td>0.029</td><td></td><td>1.29</td></tr><tr><td>grAl</td><td>[121]</td><td>30</td><td></td><td></td><td>6.54</td><td></td><td>260</td><td>0.031</td><td></td><td>1.77</td></tr><tr><td>grAl</td><td>[121]</td><td>30</td><td></td><td></td><td>5.95</td><td></td><td>180</td><td>0.049</td><td></td><td>2.76</td></tr><tr><td>grAl Transmon</td><td>[78]</td><td>10</td><td>1800 (RT)</td><td>1.9</td><td>7.57*</td><td></td><td>1100</td><td></td><td></td><td>3.96*</td></tr><tr><td>grAl Transmon</td><td>[79]</td><td>20</td><td>7000*</td><td>1.8</td><td>5.47</td><td></td><td>270*</td><td></td><td></td><td>1.16</td></tr><tr><td>grAl Transmon</td><td>[79]</td><td>20</td><td>7000*</td><td>1.8</td><td>8.50</td><td></td><td>1891*</td><td></td><td></td><td>2.13</td></tr><tr><td>grAl Transmon</td><td>[79]</td><td>20</td><td>10000*</td><td>1.8</td><td>7.93</td><td></td><td>1350*</td><td></td><td></td><td>1.69</td></tr><tr><td>Hf</td><td>[122]</td><td>125</td><td>7.3</td><td>0.385</td><td>5*</td><td></td><td>15.3</td><td>0.16</td><td></td><td>3.25</td></tr><tr><td>Hf</td><td>[122]</td><td>125</td><td>7.6</td><td>0.364</td><td>5*</td><td></td><td>17</td><td>0.15</td><td></td><td>1.74</td></tr><tr><td>Hf</td><td>[122]</td><td>125</td><td>8.25</td><td>0.403</td><td>5*</td><td></td><td>16.5</td><td>0.15</td><td></td><td>4.05</td></tr><tr><td>Hf</td><td>[122]</td><td>125</td><td>10.2</td><td>0.410</td><td>5*</td><td></td><td>20.6</td><td>0.12</td><td></td><td>6.05</td></tr><tr><td>Hf</td><td>[122]</td><td>125</td><td>8.35</td><td>0.435</td><td>5*</td><td></td><td>15.8</td><td>0.16</td><td></td><td>5.15</td></tr><tr><td>Hf</td><td>[122]</td><td>125</td><td>6.46</td><td>0.355</td><td>5*</td><td></td><td>13.2</td><td>0.19</td><td></td><td>0.16</td></tr><tr><td>Hf</td><td>[122]</td><td>125</td><td>8.11</td><td>0.395</td><td>5*</td><td></td><td>16.7</td><td>0.15</td><td></td><td>1.9</td></tr><tr><td>Hf</td><td>[123]</td><td>250</td><td></td><td></td><td></td><td>[123] Fig. 3</td><td>4.8</td><td></td><td></td><td>[123] Fig. 3</td></tr><tr><td>Hf</td><td>[123]</td><td>250</td><td></td><td></td><td></td><td>[123] Fig. 3</td><td>5.3</td><td></td><td></td><td>[123] Fig. 3</td></tr><tr><td>a-InO</td><td>[27] TC002</td><td>40</td><td>1450</td><td>3.2</td><td>5.89</td><td></td><td>590</td><td>11.410<sup>-3</sup></td><td>1</td><td>0.10</td></tr><tr><td>a-InO</td><td>[27] TC014</td><td>40</td><td>1680</td><td>3.16</td><td>2.75</td><td></td><td>700</td><td>10.610<sup>-3</sup></td><td>1</td><td>0.16</td></tr><tr><td>a-InO</td><td>[27] TC014</td><td>40</td><td>1680</td><td>3.16</td><td>3.74</td><td></td><td>700</td><td>15.210<sup>-3</sup></td><td>1</td><td>0.105</td></tr><tr><td>a-InO</td><td>[27] TC003</td><td>40</td><td>2060</td><td>2.8</td><td>5.08</td><td></td><td>910</td><td>8.610<sup>-3</sup></td><td>1</td><td>0.063</td></tr><tr><td>a-InO</td><td>[27] TC003</td><td>40</td><td>2060</td><td>2.8</td><td>6.13</td><td></td><td>910</td><td>7.110<sup>-3</sup></td><td>1</td><td>0.059</td></tr><tr><td>a-InO</td><td>[27] TC003</td><td>40</td><td>2060</td><td>2.8</td><td>7.14</td><td></td><td>910</td><td>6.110<sup>-3</sup></td><td>1</td><td>0.072</td></tr><tr><td>a-InO</td><td>[27] TC040</td><td>40</td><td>2840</td><td>2.74</td><td>6.21</td><td></td><td>1320</td><td>4.810<sup>-3</sup></td><td>1</td><td>0.10</td></tr><tr><td>a-InO</td><td>[27] TC040</td><td>40</td><td>2840</td><td>2.74</td><td>7.82</td><td></td><td>1320</td><td>3.810<sup>-3</sup></td><td>1</td><td>0.07</td></tr><tr><td>a-InO</td><td>[27] TC001</td><td>40</td><td>3360</td><td>2.24</td><td>6.23</td><td></td><td>1790</td><td>3.510<sup>-3</sup></td><td>1</td><td>0.12</td></tr><tr><td>a-InO</td><td>[27] TC001</td><td>40</td><td>3360</td><td>2.24</td><td>3.41</td><td></td><td>1790</td><td>6.510<sup>-3</sup></td><td>1</td><td>0.10</td></tr><tr><td>a-InO</td><td>DP-res9</td><td>40</td><td></td><td></td><td>1.66</td><td></td><td>2300</td><td>0.01</td><td>1</td><td>0.05</td></tr><tr><td>a-InO</td><td>[27] TC007-2</td><td>40</td><td>5950</td><td>1.6</td><td>3.98</td><td></td><td>4060</td><td>2.410<sup>-3</sup></td><td>1</td><td>0.088</td></tr><tr><td>a-InO</td><td>[27] TC007-1</td><td>40</td><td>7470</td><td>1.4</td><td>3.86</td><td></td><td>5680</td><td>1.810<sup>-3</sup></td><td>1</td><td>0.079</td></tr><tr><td>a-InO</td><td>[27] TC016-8</td><td>40</td><td>12100</td><td>0.67</td><td>3.56</td><td></td><td>11640</td><td>0.9610<sup>-3</sup></td><td>1</td><td>0.043</td></tr><tr><td>a-InO</td><td>[27] TC017</td><td>40</td><td>14250</td><td>0.49</td><td>3.79</td><td></td><td>10650</td><td>0.9810<sup>-3</sup></td><td>1</td><td>0.03</td></tr><tr><td>a-InO</td><td>[27] TC016-6</td><td>40</td><td>15950</td><td>0.47</td><td>2.77</td><td></td><td>16680</td><td>0.8610<sup>-3</sup></td><td>1</td><td>0.025</td></tr><tr><td>Nb</td><td>[32]</td><td>100</td><td>0.293</td><td>9.02</td><td>4.137</td><td>60</td><td></td><td>85</td><td></td><td>32.62</td></tr><tr><td>Nb</td><td>[32]</td><td>100</td><td>0.293</td><td>9.02</td><td>4.787</td><td>60</td><td></td><td>73</td><td></td><td>40.22</td></tr><tr><td>Nb</td><td>[32]</td><td>100</td><td>0.293</td><td>9.02</td><td>5.367</td><td>60</td><td></td><td>65</td><td></td><td>42.80</td></tr><tr><td>Nb</td><td>[32]</td><td>100</td><td>0.293</td><td>9.02</td><td>4.503</td><td>60</td><td></td><td>78</td><td></td><td>47.25</td></tr><tr><td>Nb</td><td>[32]</td><td>100</td><td>0.293</td><td>9.02</td><td>3.868</td><td>60</td><td></td><td>91</td><td></td><td>24.31</td></tr><tr><td>Nb</td><td>[32]</td><td>100</td><td>0.293</td><td>9.02</td><td>5.534</td><td>60</td><td></td><td>63</td><td></td><td>49.44</td></tr><tr><td>Nb</td><td>[124]</td><td>150</td><td>0.23</td><td>9.3</td><td>6.417</td><td>64</td><td></td><td>48</td><td>1.210<sup>-2</sup></td><td>4</td></tr><tr><td>Nb</td><td>[125]</td><td>180</td><td></td><td></td><td>6.5</td><td>64</td><td></td><td>47.5</td><td></td><td>6</td></tr><tr><td>Nb</td><td>[126]</td><td>190</td><td>0.0147</td><td>9.4</td><td>4.887</td><td>32 [127]</td><td></td><td>253</td><td>3.810<sup>-3</sup></td><td>75</td></tr><tr><td>Nb</td><td>[128]</td><td>400</td><td>0.012</td><td>9.2</td><td>5.34</td><td>38.6</td><td></td><td>15.8</td><td>3.810<sup>-2</sup></td><td>1.5</td></tr><tr><td>Nb</td><td>[128]</td><td>400</td><td>0.012</td><td>9.2</td><td>5.4</td><td>38.6</td><td></td><td>15.7</td><td>1.910<sup>-2</sup></td><td>3.1</td></tr><tr><td>Nb</td><td>[128]</td><td>400</td><td>0.012</td><td>9.2</td><td>5.56</td><td>38.6</td><td></td><td>15.2</td><td>1.210<sup>-2</sup></td><td>3.6</td></tr><tr><td>Nb</td><td>[128]</td><td>400</td><td>0.012</td><td>9.2</td><td>5.73</td><td>38.6</td><td></td><td>14.7</td><td>7.510<sup>-3</sup></td><td>5.6</td></tr><tr><td>Nb</td><td>[128]</td><td>400</td><td>0.012</td><td>9.2</td><td>5.93</td><td>38.6</td><td></td><td>14.3</td><td>5.310<sup>-3</sup></td><td>6.7</td></tr><tr><td>Nb</td><td>[129]</td><td>185</td><td>0.21</td><td></td><td>7</td><td>70.4</td><td></td><td>36.5</td><td>210<sup>-2</sup></td><td>3.4</td></tr><tr><td>Nb</td><td>[129]</td><td>185</td><td>0.21</td><td></td><td>7.5</td><td>70.4</td><td></td><td>34</td><td>210<sup>-2</sup></td><td>2.5</td></tr></tbody></table>

Continued on next page
<table><caption>Supplementary Materials Table I – Continued from previous page</caption><thead><tr><th>Material</th><th>Ref</th><th>d (nm)</th><th>R<sub>□</sub> (Ω/□)</th><th>T<sub>c</sub> (K)</th><th>f (GHz)</th><th>λ (nm)</th><th>L<sub>K</sub> (pH/□)</th><th>σ<sub>2</sub> (×10<sup>8</sup> S/m)</th><th>α</th><th>Q<sub>i</sub> (×10<sup>5</sup>)</th></tr></thead><tbody><tr><td>Nb</td><td>[129]</td><td>185</td><td>0.21</td><td></td><td>8</td><td>70.4</td><td></td><td>31.9</td><td>2 10<sup>-2</sup></td><td>2.3</td></tr><tr><td>Nb</td><td>[129]</td><td>185</td><td>0.21</td><td></td><td>8.5</td><td>70.4</td><td></td><td>30</td><td>2 10<sup>-2</sup></td><td>1.8</td></tr><tr><td>Nb</td><td>[129]</td><td>185</td><td>0.12</td><td></td><td>7</td><td>57.3</td><td></td><td>55</td><td>1.6 10<sup>-2</sup></td><td>9.3</td></tr><tr><td>Nb</td><td>[129]</td><td>185</td><td>0.12</td><td></td><td>7.5</td><td>57.3</td><td></td><td>51.4</td><td>1.6 10<sup>-2</sup></td><td>7.2</td></tr><tr><td>Nb</td><td>[129]</td><td>185</td><td>0.12</td><td></td><td>8</td><td>57.3</td><td></td><td>48.2</td><td>1.6 10<sup>-2</sup></td><td>10.7</td></tr><tr><td>Nb/Au</td><td>[130]</td><td>112</td><td></td><td></td><td>1.753</td><td></td><td>0.24</td><td>33.5</td><td>0.1</td><td>14.8</td></tr><tr><td>Nb Transmon</td><td>[81]</td><td>120</td><td>0.22</td><td>9.1</td><td>Tab. I</td><td>57</td><td>0.033</td><td></td><td></td><td>Tab. I</td></tr><tr><td>Nb Transmon</td><td>[131]</td><td></td><td></td><td></td><td>SI Tab. I,II</td><td>60*</td><td></td><td></td><td></td><td>SI Tab. I,II</td></tr><tr><td>Nb Transmon</td><td>[109]</td><td>200</td><td></td><td></td><td>5*</td><td>~ 100*</td><td></td><td></td><td></td><td>Fig. 2b</td></tr><tr><td>Nb Transmon</td><td>[5]</td><td>200</td><td></td><td></td><td>2.89</td><td>40</td><td></td><td></td><td></td><td>91</td></tr><tr><td>Nb Transmon</td><td>[5]</td><td>200</td><td></td><td></td><td>2.89</td><td>40</td><td></td><td></td><td></td><td>138.9</td></tr><tr><td>Nb thick</td><td>[132]</td><td>6100</td><td>1.3 10<sup>-3</sup></td><td></td><td>1.3</td><td>42.5</td><td></td><td>538</td><td>1.6 10<sup>-6</sup></td><td>3 10<sup>4</sup></td></tr><tr><td>Nb (3D)</td><td>[133]</td><td>bulk</td><td></td><td></td><td>1.3</td><td>32 [127]</td><td></td><td>951</td><td>1.2 10<sup>-6</sup></td><td>2.3 10<sup>4</sup></td></tr><tr><td>Nb (3D)</td><td>[133]</td><td>bulk</td><td></td><td></td><td>1.3</td><td>32 [127]</td><td></td><td>951</td><td>1.2 10<sup>-6</sup></td><td>15 10<sup>4</sup></td></tr><tr><td>Nb (3D)</td><td>[133]</td><td>bulk</td><td></td><td></td><td>2.6</td><td>32 [127]</td><td></td><td>475</td><td>2.5 10<sup>-6</sup></td><td>1.5 10<sup>4</sup></td></tr><tr><td>Nb (3D)</td><td>[133]</td><td>bulk</td><td></td><td></td><td>5.0</td><td>32 [127]</td><td></td><td>247</td><td>4.8 10<sup>-6</sup></td><td>1.0 10<sup>4</sup></td></tr><tr><td>Nb (3D)</td><td>[133]</td><td>bulk</td><td></td><td></td><td>5.0</td><td>32 [127]</td><td></td><td>247</td><td>4.8 10<sup>-6</sup></td><td>15 10<sup>4</sup></td></tr><tr><td>Nb (3D)</td><td>[134]</td><td>bulk</td><td></td><td></td><td>6.6</td><td>32 [127]</td><td></td><td>187</td><td></td><td>1.5 10<sup>4</sup></td></tr><tr><td>Nb (3D)</td><td>[135]</td><td>bulk</td><td></td><td></td><td>7.9</td><td>39.6</td><td></td><td>102</td><td>3.3 10<sup>-5</sup></td><td>0.5 10<sup>4</sup></td></tr><tr><td>Nb (3D)</td><td>[136]</td><td>bulk</td><td></td><td></td><td>4.3</td><td>33</td><td></td><td>270</td><td>5.3 10<sup>-6</sup></td><td>3 10<sup>4</sup></td></tr><tr><td>Nb (3D)</td><td>[137]</td><td>bulk</td><td></td><td></td><td>6.5*</td><td>33</td><td></td><td>179</td><td></td><td>[137] Fig. 1</td></tr><tr><td>Nb (3D)</td><td>[138]</td><td>bulk</td><td></td><td></td><td>98.2</td><td>40</td><td></td><td>8.06</td><td></td><td>300</td></tr><tr><td>NbN</td><td>[139]</td><td>13</td><td>106</td><td>5</td><td>5.58</td><td></td><td></td><td>34.5</td><td>0.6</td><td>1</td><td>0.861</td></tr><tr><td>NbN</td><td>[139]</td><td>13</td><td>122</td><td>5.2</td><td>5.87</td><td></td><td></td><td>38.2</td><td>0.5</td><td>1</td><td>0.753</td></tr><tr><td>NbN</td><td>[139]</td><td>13</td><td>196</td><td>7.5</td><td>5.95</td><td></td><td></td><td>44.4</td><td>0.5</td><td>1</td><td>0.885</td></tr><tr><td>NbN</td><td>[139]</td><td>13</td><td>225</td><td>6.5</td><td>5.49</td><td></td><td></td><td>52.5</td><td>0.4</td><td>1</td><td>0.59</td></tr><tr><td>NbN</td><td>[139]</td><td>13</td><td>267</td><td>6</td><td>5.72</td><td></td><td></td><td>57.2</td><td>0.35</td><td>1</td><td>0.768</td></tr><tr><td>NbN</td><td>[139]</td><td>13</td><td>310</td><td>5.8</td><td>4.93</td><td></td><td></td><td>76.8</td><td>0.3</td><td>1</td><td>0.463</td></tr><tr><td>NbN</td><td>[139]</td><td>13</td><td>362</td><td>5.6</td><td>6.47</td><td></td><td></td><td>91.3</td><td>0.2</td><td>1</td><td>0.993</td></tr><tr><td>NbN</td><td>[139]</td><td>13</td><td>518</td><td>4.2</td><td>3.34</td><td></td><td></td><td>173.3</td><td>0.2</td><td>1</td><td>0.176</td></tr><tr><td>NbN</td><td>[140]</td><td>20</td><td>503</td><td>7.2</td><td>4.835</td><td></td><td></td><td>83</td><td>0.2</td><td>1</td><td>0.25</td></tr><tr><td>NbN</td><td>[141]</td><td>100</td><td></td><td></td><td>5.952</td><td></td><td></td><td>0.178</td><td>15</td><td>0.1</td><td>0.957</td></tr><tr><td>NbN</td><td>[142]</td><td>10</td><td>1033</td><td>7.4</td><td>5.7</td><td></td><td></td><td>192</td><td>0.14</td><td>0.94</td><td>0.3</td></tr><tr><td>NbN</td><td>[142]</td><td>10</td><td>1033</td><td>7.4</td><td>5.756</td><td></td><td></td><td>192</td><td>0.14</td><td>1</td><td>1.58</td></tr><tr><td>NbN</td><td>[142]</td><td>10</td><td>1033</td><td>7.4</td><td>5.56</td><td></td><td></td><td>192</td><td>0.15</td><td>1</td><td>3.24</td></tr><tr><td>NbN</td><td>[143]</td><td>6</td><td>959</td><td>8.5</td><td>6.835</td><td></td><td></td><td>132</td><td>0.3</td><td>1</td><td>0.75</td></tr><tr><td>NbN</td><td>[120]</td><td>13</td><td></td><td></td><td>[120] Tab. II</td><td></td><td></td><td>89</td><td></td><td></td><td>[120] Tab. II</td></tr><tr><td>NbN Transmon</td><td>[144]</td><td>8</td><td></td><td></td><td>Fig. 7b</td><td></td><td></td><td>200</td><td></td><td></td><td>Fig. 7b</td></tr><tr><td>NbN Transmon</td><td>[109]</td><td>400</td><td></td><td></td><td>5*</td><td>~ 300*</td><td></td><td>~ 0.3*</td><td></td><td></td><td>Fig. 2b</td></tr><tr><td>NbSi</td><td>[145]</td><td>15</td><td>600</td><td>0.85</td><td>6.21</td><td></td><td></td><td>830</td><td>2.05 10<sup>-2</sup></td><td></td><td>0.019</td></tr><tr><td>NbSi</td><td>[145]</td><td>15</td><td>600</td><td>0.85</td><td>6.44</td><td></td><td></td><td>830</td><td>1.98 10<sup>-2</sup></td><td></td><td>0.023</td></tr><tr><td>NbSi</td><td>[145]</td><td>15</td><td>600</td><td>0.85</td><td>6.73</td><td></td><td></td><td>830</td><td>1.89 10<sup>-2</sup></td><td></td><td>0.020</td></tr><tr><td>NbSi</td><td>[145]</td><td>15</td><td>600</td><td>0.85</td><td>7.00</td><td></td><td></td><td>830</td><td>1.82 10<sup>-2</sup></td><td></td><td>0.021</td></tr><tr><td>NbSi</td><td>[145]</td><td>15</td><td>600</td><td>0.85</td><td>7.29</td><td></td><td></td><td>830</td><td>1.75 10<sup>-2</sup></td><td></td><td>0.026</td></tr><tr><td>NbTiN</td><td>[146]</td><td>8</td><td>250 (RT)</td><td>9.3</td><td>2.77</td><td></td><td></td><td>35</td><td>2</td><td></td><td>2.11</td></tr><tr><td>NbTiN</td><td>[146]</td><td>8</td><td>250 (RT)</td><td>9.3</td><td>2.98</td><td></td><td></td><td>35</td><td>1.9</td><td></td><td>2.18</td></tr><tr><td>NbTiN</td><td>[146]</td><td>8</td><td>250 (RT)</td><td>9.3</td><td>3.72</td><td></td><td></td><td>35</td><td>1.5</td><td></td><td>1.32</td></tr><tr><td>NbTiN</td><td>[146]</td><td>8</td><td>250 (RT)</td><td>9.3</td><td>4.06</td><td></td><td></td><td>35</td><td>1.4</td><td></td><td>1.46</td></tr><tr><td>NbTiN</td><td>[146]</td><td>8</td><td>250 (RT)</td><td>9.3</td><td>4.54</td><td></td><td></td><td>35</td><td>1.25</td><td></td><td>2.28</td></tr><tr><td>NbTiN</td><td>[147]</td><td>300</td><td>3.66 (RT?)</td><td>15.5</td><td>2.75</td><td></td><td></td><td>0.32</td><td>6</td><td></td><td>20.9</td></tr><tr><td>NbTiN</td><td>[147]</td><td>300</td><td>3.66 (RT?)</td><td>15.5</td><td>3.68</td><td></td><td></td><td>0.32</td><td>4.5</td><td></td><td>18.3</td></tr><tr><td>NbTiN</td><td>[147]</td><td>300</td><td>3.66 (RT?)</td><td>15.5</td><td>4.57</td><td></td><td></td><td>0.32</td><td>3.6</td><td></td><td>13.5</td></tr><tr><td>NbTiN</td><td>[147]</td><td>300</td><td>3.66 (RT?)</td><td>15.5</td><td>6.41</td><td></td><td></td><td>0.32</td><td>2.6</td><td></td><td>20.2</td></tr><tr><td>NbTiN</td><td>[148]</td><td>5</td><td>817</td><td>9.5</td><td>2.5*</td><td></td><td></td><td>118.6</td><td>1</td><td></td><td>0.3</td></tr><tr><td>NbTiN</td><td>[148]</td><td>10</td><td>277</td><td>11.6</td><td>2*</td><td></td><td></td><td>32.9</td><td>2.4</td><td></td><td>0.5</td></tr><tr><td>NbTiN</td><td>[148]</td><td>20</td><td>100</td><td>13.1</td><td>3*</td><td></td><td></td><td>10.5</td><td>2.5</td><td></td><td>0.7</td></tr><tr><td>NbTiN</td><td>[149]</td><td>12</td><td></td><td></td><td>7.03</td><td></td><td></td><td>1.248</td><td></td><td></td><td>0.14</td></tr></tbody></table>

Continued on next page
Supplementary Materials Table I – Continued from previous page

<table><thead><tr><th>Material</th><th>Ref</th><th>d (nm)</th><th>R<sub>□</sub> (Ω/□)</th><th>T<sub>c</sub> (K)</th><th>f (GHz)</th><th>λ (nm)</th><th>L<sub>K</sub> (pH/□)</th><th>σ<sub>2</sub> (×10<sup>8</sup> S/m)</th><th>α</th><th>Q<sub>i</sub> (×10<sup>5</sup>)</th></tr></thead><tbody><tr><td>NbTiN</td><td>[149]</td><td>12</td><td></td><td></td><td>8.08</td><td></td><td>0.965</td><td></td><td></td><td>0.11</td></tr><tr><td>NbTiN</td><td>[149]</td><td>12</td><td></td><td></td><td>7.02</td><td></td><td>1.252</td><td></td><td></td><td>0.21</td></tr><tr><td>NbTiN</td><td>[150]</td><td>10</td><td></td><td></td><td>[150] Tab. I</td><td></td><td>[150] Tab. I</td><td></td><td></td><td>[150] Tab. I</td></tr><tr><td>PtSi</td><td>[151]</td><td>60</td><td></td><td></td><td>4.876</td><td></td><td>8.2</td><td>0.66</td><td></td><td>1.47</td></tr><tr><td>Re</td><td>[152]</td><td>25</td><td>1.905</td><td>1.85</td><td>6.034</td><td></td><td>1.42</td><td>7.5</td><td></td><td>0.14</td></tr><tr><td>Re</td><td>[152]</td><td>100</td><td>0.054</td><td>1.88</td><td>6.99</td><td></td><td>0.04</td><td>57</td><td></td><td>0.08</td></tr><tr><td>α-Ta</td><td>[33]</td><td>200</td><td></td><td></td><td>4.484</td><td></td><td>0.063*</td><td>28</td><td></td><td>95</td></tr><tr><td>α-Ta</td><td>[153]</td><td>100</td><td>1.0</td><td>2.9</td><td>6*</td><td></td><td>0.476</td><td>5.5</td><td></td><td>5.5</td></tr><tr><td>α-Ta</td><td>[153]</td><td>100</td><td>1.0</td><td>2.9</td><td>6*</td><td></td><td>0.476</td><td>5.5</td><td></td><td>11.5</td></tr><tr><td>α-Ta</td><td>[153]</td><td>100</td><td>1.0</td><td>2.9</td><td>6*</td><td></td><td>0.476</td><td>5.5</td><td></td><td>7</td></tr><tr><td>α-Ta</td><td>[153]</td><td>100</td><td>1.0</td><td>2.9</td><td>6*</td><td></td><td>0.476</td><td>5.5</td><td></td><td>8.8</td></tr><tr><td>α-Ta</td><td>[153]</td><td>100</td><td>1.0</td><td>2.9</td><td>6*</td><td></td><td>0.476</td><td>5.5</td><td></td><td>12.8</td></tr><tr><td>α-Ta</td><td>[153]</td><td>100</td><td>1.0</td><td>2.9</td><td>6*</td><td></td><td>0.476</td><td>5.5</td><td></td><td>25.1</td></tr><tr><td>α-Ta</td><td>[153]</td><td>100</td><td>1.0</td><td>2.9</td><td>6*</td><td></td><td>0.476</td><td>5.5</td><td></td><td>10.6</td></tr><tr><td>α-Ta</td><td>[153]</td><td>100</td><td>1.0</td><td>2.9</td><td>6*</td><td></td><td>0.476</td><td>5.5</td><td></td><td>33.3</td></tr><tr><td>α-Ta</td><td>[153]</td><td>100</td><td>1.0</td><td>2.9</td><td>6*</td><td></td><td>0.476</td><td>5.5</td><td></td><td>23.7</td></tr><tr><td>α-Ta</td><td>[154]</td><td>150</td><td>0.1*</td><td>4.12</td><td>5.1</td><td></td><td>0.033*</td><td>62</td><td>7.810<sup>-3</sup></td><td>14</td></tr><tr><td>α-Ta</td><td>[154]</td><td>150</td><td>0.1*</td><td>4.12</td><td>5.4</td><td></td><td>0.033*</td><td>58.5</td><td>7.810<sup>-3</sup></td><td>14</td></tr><tr><td>α-Ta</td><td>[154]</td><td>150</td><td>0.1*</td><td>4.12</td><td>5.9</td><td></td><td>0.033*</td><td>53.5</td><td>7.810<sup>-3</sup></td><td>12.5</td></tr><tr><td>α-Ta</td><td>[154]</td><td>150</td><td>0.1*</td><td>4.12</td><td>6.2</td><td></td><td>0.033*</td><td>51</td><td>7.810<sup>-3</sup></td><td>12</td></tr><tr><td>α-Ta</td><td>[154]</td><td>150</td><td>0.1*</td><td>4.12</td><td>6.4</td><td></td><td>0.033*</td><td>49.5</td><td>7.810<sup>-3</sup></td><td>15</td></tr><tr><td>α-Ta</td><td>[154]</td><td>150</td><td>0.1*</td><td>4.12</td><td>6.8</td><td></td><td>0.033*</td><td>46.5</td><td>7.810<sup>-3</sup></td><td>10</td></tr><tr><td>α-Ta</td><td>[154]</td><td>150</td><td>0.1*</td><td>4.12</td><td>7.1</td><td></td><td>0.033*</td><td>44.5</td><td>7.810<sup>-3</sup></td><td>7.5</td></tr><tr><td>α-Ta</td><td>[155]</td><td>200</td><td>0.1</td><td>4.42</td><td>Fig. 3b</td><td></td><td>0.031</td><td>24.5 – 29</td><td>7.710<sup>-3</sup></td><td>Fig. 3b</td></tr><tr><td>α-Ta</td><td>[156]</td><td>50</td><td>0.15</td><td>4.14</td><td>5.5</td><td>45</td><td>0.071</td><td>81.3</td><td>2.810<sup>-2</sup></td><td>19</td></tr><tr><td>α-Ta</td><td>[156]</td><td>50</td><td>0.15</td><td>4.14</td><td>6.5</td><td>45</td><td>0.071</td><td>68.8</td><td>2.810<sup>-2</sup></td><td>7</td></tr><tr><td>α-Ta</td><td>[157]</td><td>300</td><td></td><td></td><td>6.63</td><td>150</td><td>0.195</td><td>8.48</td><td>2.410<sup>-2</sup></td><td>25</td></tr><tr><td>α-Ta</td><td>[158]</td><td>100</td><td>0.078</td><td>4.3</td><td>6</td><td></td><td>0.025</td><td>106</td><td></td><td>41.6</td></tr><tr><td>α-Ta</td><td>[158]</td><td>100</td><td>0.078</td><td>4.3</td><td>5.3</td><td></td><td>0.025</td><td>120</td><td></td><td>34.4</td></tr><tr><td>α-Ta</td><td>[158]</td><td>100</td><td>1.16</td><td>4.45</td><td>5.4</td><td></td><td>0.36</td><td>8.2</td><td></td><td>62.5</td></tr><tr><td>α-Ta</td><td>[158]</td><td>100</td><td>1.16</td><td>4.45</td><td>4.6</td><td></td><td>0.36</td><td>9.6</td><td></td><td>24.3</td></tr><tr><td>α-Ta</td><td>[4]</td><td>150</td><td>0.015</td><td>4.3</td><td>[4] Tab. S5</td><td>150</td><td></td><td></td><td></td><td>[4] Tab. S1</td></tr><tr><td>α-Ta Transmon</td><td>[81]</td><td>120</td><td>0.24</td><td>4.2</td><td>Tab. I</td><td></td><td>0.08</td><td></td><td></td><td>Tab. I</td></tr><tr><td>α-Ta Transmon</td><td>[80]</td><td>200</td><td>0.2</td><td>4.38</td><td>SI Tab. I</td><td></td><td>0.063</td><td></td><td></td><td>SI Tab. I</td></tr><tr><td>α-Ta Transmon</td><td>[7]</td><td>200*</td><td>0.27*</td><td>4.2</td><td>[7] Tab. I</td><td>150*</td><td></td><td></td><td></td><td>[7] Tab. I</td></tr><tr><td>β-Ta</td><td>[159]</td><td>60</td><td>39.8</td><td>1.0</td><td>8*</td><td></td><td>54.6</td><td>0.060</td><td></td><td>4.3</td></tr><tr><td>TiAlV (3D)</td><td>[160]</td><td>bulk</td><td></td><td>4.5</td><td>7.5</td><td>8000</td><td></td><td>2.6310<sup>-3</sup></td><td>410<sup>-3</sup></td><td>0.3</td></tr><tr><td>TiAlV (3D)</td><td>[161]</td><td>bulk</td><td></td><td>3.5</td><td>5.917</td><td>657</td><td>0.82</td><td>0.49</td><td></td><td>22</td></tr><tr><td>TiN</td><td>[26]</td><td>3</td><td>522</td><td>3</td><td>5.563</td><td></td><td>239</td><td>0.4</td><td>0.99</td><td>0.9</td></tr><tr><td>TiN</td><td>[94]</td><td>60</td><td>3.23</td><td>5.4</td><td>5.5*</td><td></td><td>0.82</td><td>5.9</td><td>0.25</td><td>0.95</td></tr><tr><td>TiN</td><td>[94]</td><td>60</td><td>3.23</td><td>5.4</td><td>5.5*</td><td></td><td>0.82</td><td>5.9</td><td>0.25</td><td>0.72</td></tr><tr><td>TiN</td><td>[94]</td><td>60</td><td>3.23</td><td>5.4</td><td>5.5*</td><td></td><td>0.82</td><td>5.9</td><td>0.25</td><td>0.61</td></tr><tr><td>TiN</td><td>[94]</td><td>60</td><td>3.23</td><td>5.4</td><td>5.5*</td><td></td><td>0.82</td><td>5.9</td><td>0.25</td><td>0.54</td></tr><tr><td>TiN</td><td>[94]</td><td>60</td><td>3.23</td><td>5.4</td><td>5.5*</td><td></td><td>0.82</td><td>5.9</td><td>0.25</td><td>0.51</td></tr><tr><td>TiN</td><td>[97]</td><td>100</td><td>9</td><td>3.8</td><td>6*</td><td></td><td>3.27</td><td>0.8</td><td>0.43</td><td>3.3</td></tr><tr><td>TiN</td><td>[97]</td><td>100</td><td>9</td><td>3.8</td><td>6*</td><td></td><td>3.27</td><td>0.8</td><td>0.43</td><td>1.8</td></tr><tr><td>TiN</td><td>[162]</td><td>8.9</td><td>505 (RT)</td><td>3.01</td><td>5*</td><td></td><td>234</td><td>0.15</td><td></td><td>[162] Fig.2c</td></tr><tr><td>TiN</td><td>[162]</td><td>14.2</td><td>145 (RT)</td><td>3.63</td><td>5*</td><td></td><td>56</td><td>0.4</td><td></td><td>[162] Fig.2c</td></tr><tr><td>TiN</td><td>[162]</td><td>49.8</td><td>20.7 (RT)</td><td>4.05</td><td>5*</td><td></td><td>7.1</td><td>0.9</td><td></td><td>[162] Fig.2c</td></tr><tr><td>TiN</td><td>[162]</td><td>109</td><td>5.68 (RT)</td><td>4.62</td><td>5*</td><td></td><td>1.7</td><td>1.7</td><td></td><td>[162] Fig.2c</td></tr><tr><td>TiN</td><td>[163]</td><td>100</td><td>316</td><td>2.95</td><td>10*</td><td></td><td>148</td><td>0.01</td><td></td><td>0.022</td></tr><tr><td>TiN</td><td>[164]</td><td>100</td><td>10</td><td>3.5</td><td>5*</td><td></td><td>3.94</td><td>0.80</td><td></td><td>50</td></tr><tr><td>TiN</td><td>[165]</td><td>12</td><td>361</td><td>2.8</td><td>5.025</td><td></td><td>178</td><td>0.15</td><td></td><td>39</td></tr><tr><td>TiN</td><td>[166]</td><td>35</td><td></td><td></td><td>[166] Fig. 5</td><td></td><td>[166] Tab. 2</td><td></td><td></td><td>[166] Fig. 5</td></tr><tr><td>TiN Transmon</td><td>[45]</td><td>100</td><td>9</td><td>3.8</td><td>Tab. I</td><td></td><td>3.27 [97]</td><td></td><td></td><td>Tab. I</td></tr><tr><td>TiN/Al</td><td>[167]</td><td>100</td><td>0.1323</td><td>3.2</td><td>4.3733</td><td></td><td>57</td><td>0.06</td><td>0.88</td><td>2.7</td></tr></tbody></table>

Continued on next page
Supplementary Materials Table I – Continued from previous page

<table><thead><tr><th>Material</th><th>Ref</th><th>d (nm)</th><th>R<sub>□</sub> (Ω/□)</th><th>T<sub>c</sub> (K)</th><th>f (GHz)</th><th>λ (nm)</th><th>L<sub>K</sub> (pH/□)</th><th>σ<sub>2</sub> (×10<sup>8</sup> S/m)</th><th>α</th><th>Q<sub>i</sub> (×10<sup>5</sup>)</th></tr></thead><tbody><tr><td>TiN/Al</td><td>[168]</td><td>10</td><td></td><td></td><td>4.300</td><td></td><td>1200</td><td>3.08 10<sup>-2</sup></td><td>1</td><td>4.54</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>10</td><td></td><td></td><td>4.370</td><td></td><td>1200</td><td>3.03 10<sup>-2</sup></td><td>1</td><td>4.47</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>10</td><td></td><td></td><td>4.407</td><td></td><td>1200</td><td>3.00 10<sup>-2</sup></td><td>1</td><td>3.61</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>10</td><td></td><td></td><td>4.418</td><td></td><td>1200</td><td>3.00 10<sup>-2</sup></td><td>1</td><td>2.61</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>10</td><td></td><td></td><td>4.500</td><td></td><td>1200</td><td>2.94 10<sup>-2</sup></td><td>1</td><td>4.11</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>10</td><td></td><td></td><td>4.537</td><td></td><td>1200</td><td>2.92 10<sup>-2</sup></td><td>1</td><td>3.07</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>10</td><td></td><td></td><td>4.547</td><td></td><td>1200</td><td>2.91 10<sup>-2</sup></td><td>1</td><td>2.96</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>20</td><td></td><td></td><td>7.298</td><td></td><td>6500</td><td>1.67 10<sup>-3</sup></td><td>1</td><td>0.187</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>20</td><td></td><td></td><td>7.529</td><td></td><td>6500</td><td>1.62 10<sup>-3</sup></td><td>1</td><td>0.145</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>20</td><td></td><td></td><td>7.697</td><td></td><td>6500</td><td>1.59 10<sup>-3</sup></td><td>1</td><td>0.160</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>20</td><td></td><td></td><td>7.819</td><td></td><td>6500</td><td>1.56 10<sup>-3</sup></td><td>1</td><td>0.155</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>20</td><td></td><td></td><td>7.896</td><td></td><td>6500</td><td>1.55 10<sup>-3</sup></td><td>1</td><td>0.156</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>20</td><td></td><td></td><td>7.948</td><td></td><td>6500</td><td>1.54 10<sup>-3</sup></td><td>1</td><td>0.177</td></tr><tr><td>TiN/Al</td><td>[168]</td><td>20</td><td></td><td></td><td>8.064</td><td></td><td>6500</td><td>1.51 10<sup>-3</sup></td><td>1</td><td>0.161</td></tr><tr><td>WSi</td><td>[169]</td><td>100</td><td>20</td><td>4</td><td>1.766</td><td></td><td>6.9</td><td>1.3</td><td>0.92</td><td>7.1</td></tr><tr><td>WSi</td><td>[169]</td><td>365</td><td>12.32</td><td>1.8</td><td>4.705</td><td></td><td>9.46</td><td>0.09</td><td>0.42</td><td>1.7</td></tr><tr><td>WSi</td><td>[170]</td><td>30</td><td>98</td><td>4.75</td><td>1.032</td><td></td><td>30</td><td>1.7</td><td>0.88</td><td>1.4</td></tr><tr><td>WSi</td><td>[28]</td><td>3</td><td></td><td></td><td></td><td>[28]Fig. 1d</td><td>300</td><td></td><td></td><td>[28]Fig. 1d</td></tr><tr><td>WSi</td><td>[28]</td><td>10</td><td></td><td></td><td></td><td>[28]Fig. 1d</td><td>100</td><td></td><td></td><td>[28]Fig. 1d</td></tr></tbody></table>