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Low loss lumped-element inductors made from granular aluminum

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Granular aluminum can form compact, linear, low-loss inductors for superconducting quantum circuits, with internal quality factors exceeding three million.

desk verdict Solid experimental advance: grAl lumped-element inductors with Q above 3e6 and a clean resistivity-loss trend, but the surface-loss claim is an upper bound, not a measured equality. read the letter →

arxiv 2411.12611 v1 pith:FMVA5H4F submitted 2024-11-19 quant-ph cond-mat.mtrl-sci

Vishakha Gupta , Patrick Winkel , Neel Thakur , Peter van Vlaanderen , Yanhao Wang , Suhas Ganjam , Luigi Frunzio , Robert J. Schoelkopf This is my paper · ORCID
classification quant-phcond-mat.mtrl-sci
keywords granularaluminumkineticinductancelumped-elementresonatorssuperconductingquantumcircuitsinternalqualityfactorself-Kerrnonlinearityquasiparticledynamicshybridsuperconductorintegration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Granular aluminum is a superconductor whose microstructure gives it a large kinetic inductance, and this paper argues that this property can be turned into a practical lumped-element inductor for quantum circuits. The authors build microwave resonators in which a narrow strip of granular aluminum provides almost all of the inductance, and they show that these strips are simultaneously compact, highly linear, and low-loss. Their best all-granular-aluminum devices reach single-photon internal quality factors around $3.5 \times 10^6$, and hybrid devices with tantalum capacitors exceed $4.5 \times 10^6$, comparable to the best superconducting circuits. They also find a systematic trade-off: higher room-temperature resistivity gives more compact inductors but lower quality factors, and their analysis attributes the low-resistivity surface loss to a level similar to pure aluminum.

What carries the argument

The argument runs through the kinetic inductance of granular aluminum and its description as a one-dimensional array of effective Josephson junctions. In the device, a narrow granular-aluminum strip carries more than 90 percent of the total inductance, so the resonator frequency is set by the sheet inductance times the number of squares; the self-Kerr coefficient is diluted roughly as $E_c/N_{\text{JJ}}^2$, which lets the authors infer an effective junction size of 10 to 30 nm from measured nonlinearities. For loss, the paper uses an energy-participation-ratio budget, $1/Q_{\text{int}} = 1/Q_{\text{bulk}} + 1/Q_{\text{surf}} + 1/Q_{\text{ind}} + 1/Q_{\text{contact}}$, with the substrate bulk term and surface participation computed by finite-element simulation, and the contact term relevant only for the hybrid devices.

What would settle it

Fabricate the same resonator layouts on sapphire substrates whose bulk dielectric loss is independently varied or measured, for example annealed versus unannealed wafers with known different loss tangents, and check whether the extracted residual loss of the granular-aluminum strip stays constant. If the residual-loss-versus-resistivity trend persists unchanged under a different substrate loss, the conductor-loss attribution is supported; if the trend tracks the substrate instead, the central loss conclusion is wrong.

Watch

Extended reading notes

Core claim

The central claim is that granular aluminum, in films about 90 nm thick, can serve as a linear inductive element whose performance is no longer the limiting factor in high-coherence circuits. The measured sheet inductance ranges from 30 to 320 pH/sq, so a few-nH inductor can be made in a strip only tens of micrometers long, up to 100 times more compact than a pure-aluminum geometric inductor; the self-Kerr nonlinearity stays at 0.2 to 20 Hz per photon, far below the linewidth. Internal quality factors at single-photon power reach $3.5 \times 10^6$ for all-granular-aluminum resonators and exceed $4.5 \times 10^6$ for hybrid granular-aluminum/tantalum devices. After subtracting the known bulk dielectric loss of the substrate, the remaining loss in the lowest-resistivity films is comparable to that of standard aluminum transmon circuits, and the increase of loss with film resistivity is attributed to conductor loss in the granular aluminum rather than to a worse surface dielectric. The paper also reports that quasiparticle relaxation after a high-energy impact occurs on millisecond timescales, matching aluminum transmons and contradicting earlier thin-film granular-aluminum results.

Load-bearing premise

The load-bearing premise is that the unannealed sapphire substrates have the bulk dielectric loss tangent measured previously, $(26.6 \pm 6.9) \times 10^{-8}$; because the best measured quality factors sit near the resulting bulk limit, a different substrate loss would change the attribution of the remaining loss and weaken the conclusion that low-resistivity granular-aluminum surface loss matches pure aluminum.

Editorial extensions

If this is right

  • Granular-aluminum lumped-element inductors can serve as linear shunts in inductively shunted qubit designs without the footprint of geometric inductors.
  • Compactness reduces the number of low-frequency parasitic modes compared with millimeter-long geometric inductors.
  • Ex-situ bandage integration means granular-aluminum strips can be added to circuits whose capacitors are made from aluminum or tantalum with no measured increase in loss.
  • Resistivity becomes a design knob with a known trade-off: lower resistivity gives higher internal quality factor and lower sheet inductance.
  • At low resistivity, granular-aluminum surface loss is comparable to pure aluminum, so replacing aluminum with granular aluminum need not introduce extra surface dielectric losses.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the resistivity-loss trend is a general property of granular aluminum, coherence-critical qubit designs should favor low-resistivity strips even at the cost of area, while high-resistivity strips remain attractive for high-impedance applications such as spin-qubit readout.
  • The ex-situ bandage contact is demonstrated only with aluminum and tantalum electrodes, but the same method may extend to other high-gap superconductors; that extension is not established by this paper.
  • Because the inferred effective junction size is 10 to 30 nm, pushing strip dimensions toward that scale should eventually reveal single-junction nonlinear behavior, a regime the paper's model does not address.
  • The fast millisecond quasiparticle relaxation, if reproduced in full qubit devices, would shorten the duration of radiation-induced frequency excursions compared with earlier thin-film granular-aluminum results.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The manuscript reports the design, fabrication, and microwave characterization of lumped-element resonators whose inductance is dominated by the kinetic inductance of granular aluminum (grAl) films. The authors measure sheet inductances of 30–320 pH/sq, self-Kerr coefficients of 0.2–20 Hz/photon, and single-photon internal quality factors up to 3.5×10^6 for all-grAl devices and 4.7×10^6 for hybrid grAl/Ta devices. They also demonstrate ex-situ integration with Al and Ta capacitor electrodes using an Al bandage layer, study quasiparticle dynamics after high-energy impacts, and propose a loss budget separating bulk, surface, conductor, and contact losses. The central claim is that grAl inductors can be simultaneously compact, linear, low-loss, and compatible with standard Al/Ta circuit fabrication.

Significance. If the results hold, the paper delivers a practical solution to a recognized problem: a compact, linear, low-loss inductor for circuit QED that is lithographically simple and integrable with Al and Ta. The direct measurements are of high quality: Qint is extracted from transmission fits at single-photon powers, the self-Kerr from power-dependent frequency shifts, and Lk from frequency combined with FEM simulations of the geometry. The improvement over previous grAl resonators (best reported Q ~ 3×10^5) is substantial, and the large sample set with clear fabrication details in the appendices is a strength. The main weakness is the loss attribution: several conclusions rest on subtracting a literature bulk-loss value and on unmeasured contact losses. Nevertheless, the demonstrated Qint values themselves support the primary claim of low-loss inductors and should be of considerable interest to the circuit QED community.

major comments (3)
  1. [Sec. VI, Eq. (2), Fig. 6(c)] The conclusion that low-resistivity grAl has a surface loss factor similar to pure Al is not supported by the data as stated. For the best all-grAl device (GH23, rho_n = 555 uOhm cm, Qint = 3.66e6), subtracting the literature bulk loss of unannealed EFG sapphire (Gamma_bulk = (26.6 +/- 6.9)e-8, Qbulk ~ 4.5e6) gives a residual loss of about 5.1e-8, whereas the uncertainty in 1/Qbulk alone is about 5.8e-8. The residual is therefore consistent with zero, and the data can only place an upper bound on the grAl surface loss at this resistivity. Please propagate the bulk-loss uncertainty through Eq. (2) and rephrase the related claims in Sec. VI and the abstract (e.g., 'comparable to or lower than' instead of 'similar to').
  2. [Sec. VI, App. F, Fig. 10] The attribution of the resistivity-dependent loss to conductor loss in the grAl strip is an inference rather than a unique conclusion. The residual loss shows a saturable power dependence (App. F), which is more typical of two-level-system or quasiparticle loss than of the classical conductor loss described by Q_ind in Eq. (2). In addition, the FEM participation calculation does not include dielectric loss in the AlOx grain-boundary network, whose volume fraction and disorder increase with oxygen content (i.e., with rho_n). The authors are appropriately cautious in the abstract ('could be explained'), but Sec. VI states 'we attribute their increasing losses with resistivity to an increase in conductor loss' without ruling out these alternatives. Please soften this claim and discuss how a multimodal analysis or an explicit grain-boundary loss model could distinguish the mechanisms.
  3. [Sec. V, Sec. VI] The claim that ex-situ hybrid integration does not increase total internal losses is not backed by an independent measurement of the bandage contact loss. As the authors state in Sec. I, contact losses are not measured separately, and Sec. VI acknowledges that Q_contact may have a resistivity dependence. While the comparable Qint of hybrid and all-grAl devices from the same wafers supports the practical utility of the bandage process, the loss budget in Eq. (2) is underdetermined for the hybrids: Q_contact is degenerate with Q_ind and Q_surf. An upper-bound estimate of contact loss (e.g., from devices with different numbers of contacts or contact areas) would strengthen the integration claim.
minor comments (6)
  1. [Sec. II vs App. B] The oxygen partial pressure during grAl deposition is given as ~5e-5 mbar in Sec. II and ~5e-5 Torr in App. B; these differ by a factor of 1.333. Please harmonize the units.
  2. [Sec. III.B] The equation for the self-Kerr coefficient appears as '|K| = p2Eca2/l2 strip' due to a typesetting error; it should read K = p^2 E_c a^2 / l_strip^2.
  3. [App. E, Eq. (E1)] The term 'psurf tan δTLSp' should be 'psurf tan δ_TLS' (missing underscore and space).
  4. [Table II] The header 'W afer' should be 'Wafer'.
  5. [Fig. 3(a)] The caption uses Q_i for internal quality factor while the text uses Qint; please unify the notation.
  6. [Sec. II] 'under coupled' should be 'undercoupled'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the headline quantities are direct measurements and the loss decomposition rests on external benchmarks.

full rationale

The paper's central results—sheet inductance, self-Kerr coefficient, and single-photon internal quality factor—are directly measured. Qint and Qc are extracted by fitting the complex transmission (Eq. 1) with a circle-fit routine; Lk and Lsq are inferred from the measured resonance frequency plus FEM-simulated Lg and Cs; K is obtained from power-dependent frequency shifts. None of these quantities is defined in terms of a target conclusion. The effective-JJ model in Sec. III is used only to interpret K and to infer NJJ and Jc, and the authors explicitly state they cannot distinguish between competing microstructural pictures, so no conclusion is forced by that model. The loss analysis in Sec. VI subtracts the bulk dielectric loss of unannealed EFG sapphire using Gamma_bulk = (26.6 +/- 6.9) x 10^-8 from Refs. [10,76]. Those are prior measurements by overlapping authors, but they are independent, externally falsifiable material benchmarks, not outputs of the present fitting procedure. The grAl surface-loss comparison follows from comparing residual loss Q_res^-1 = Q_int^-1 - Q_bulk^-1 with FEM participation ratios; it is an inference, not a construction. The paper explicitly flags its limitations: it does not independently measure the bandage-contact loss, and the best-device residual loss is small relative to the uncertainty in the bulk-loss benchmark. These caveats weaken certainty in the loss attribution (correctness risk), but they do not make any step circular. No equation reduces to its own input, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The measured device performance is direct, but the interpretation of loss channels rests on external loss benchmarks and participation-ratio simulations. No new physical entities are introduced. The listed fit parameters are characterization outputs, not hidden knobs used to force the central conclusions.

free parameters (3)
  • Sheet inductance Lsq per wafer = 30-320 pH/sq
    Extracted by linear fit of Lk vs Nsq (Fig. 3b); a measured film property, but a fit parameter in the characterization.
  • TLS saturation model parameters (Q0, tanδ_TLS, nc, beta) = not tabulated per device
    Per-resonator fit used as an interpolating function to report single-photon Qint (App. E).
  • Quasiparticle recombination/trapping/generation rates = all-grAl: r=1/(16±3 ns), s=1/(1.3±0.2 ms); grAl/Ta: r=1/(30±3 ns), s=1/(2.9±0.3 ms)
    Fitted to two high-energy impact time traces (Sec. VII); reported results, not inputs to the main claim.
assumptions (6)
  • standard math The hanger transmission model S21(ω) with complex external coupling and circle fitting yields unbiased estimates of Qint and Qc.
    Used in Sec. III and App. C to extract resonator parameters; standard in cQED, not specific to grAl.
  • domain assumption Kinetic inductance fraction alpha and total Lk are obtained from measured frequency and FEM-simulated geometric inductance Lg and capacitance Cs.
    Sec. III A and App. D; the sheet inductance Lsq is then extracted from a linear fit of Lk vs Nsq. If HFSS geometry is inaccurate, Lsq and the compactness comparison shift.
  • domain assumption The loss budget separates into bulk, surface, inductor, and contact losses with known external loss factors, including bulk sapphire loss tangent (26.6±6.9)e-8 from Refs [10,76].
    Sec. VI, Eq. (2). The residual-loss and conductor-loss conclusions depend on the validity of this decomposition and on the literature values applying to these specific chips.
  • domain assumption The grAl inductor strip is modeled as a 1D array of effective Josephson junctions with homogeneous phase drop, giving K = p^2 Ec/N_JJ^2.
    Sec. III B; used to infer N_JJ, LJ, Ic, Jc. The consistency with a ~10-30 nm junction size is suggestive but not a direct microstructural verification.
  • domain assumption The power dependence of internal loss follows a generic saturable model, 1/Q(nph)=1/Q0 + psurf tanδ_TLS / (1+(nph/nc)^beta), used to interpolate single-photon Qint.
    App. E; the paper explicitly says the model is used as an interpolating function, not as a verified microscopic loss mechanism.
  • domain assumption Quasiparticle recombination-trapping-generation model dx/dt = -r x^2 - s x + g and the frequency-shift relation δf/f = -alpha/4 δx_qp describe post-impact recovery.
    Sec. VII and App. I; the millisecond relaxation times are inferred from this model. It is standard in the field but the fitted rates are not independently confirmed.

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Pith. "Pith review of Low loss lumped-element inductors made from granular aluminum." pith.science (2026). https://pith.science/paper/FMVA5H4F

@misc{pith2026241112611,
  author       = {Pith},
  title        = {Pith review of: Low loss lumped-element inductors made from granular aluminum},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FMVA5H4F}},
  note         = {Machine review of arXiv:2411.12611}
}
abstract

Lumped-element inductors are an integral component in the circuit QED toolbox. However, it is challenging to build inductors that are simultaneously compact, linear and low-loss with standard approaches that either rely on the geometric inductance of superconducting thin films or on the kinetic inductance of Josephson junctions arrays. In this work, we overcome this challenge by utilizing the high kinetic inductance offered by superconducting granular aluminum (grAl). We demonstrate lumped-element inductors with a few nH of inductance that are up to $100$ times more compact than inductors built from pure aluminum (Al). To characterize the properties of these linear inductors, we first report on the performance of lumped-element resonators built entirely out of grAl with sheet inductances varying from $30-320\,$pH/sq and self-Kerr non-linearities of $0.2-20\,\mathrm{Hz/photon}$. Further, we demonstrate ex-situ integration of these grAl inductors into hybrid resonators with Al or tantalum (Ta) capacitor electrodes without increasing total internal losses. Interestingly, the measured internal quality factors systematically decrease with increasing room-temperature resistivity of the grAl film for all devices, indicating a trade-off between compactness and internal loss. For our lowest resistivity grAl films, we measure quality factors reaching $3.5 \times 10^6$ for the all-grAl devices and $4.5 \times 10^6$ for the hybrid grAl/Ta devices, similar to state-of-the-art quantum circuits. Our loss analysis suggests that the surface loss factor of grAl is similar to that of pure Al for our lowest resistivity films, while the increasing losses with resistivity could be explained by increasing conductor loss in the grAl film.

Figures

Figures reproduced from arXiv: 2411.12611 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: a) is especially important for shorter strip lengths in order to accurately estimate NJJ (see App. D). Using this simple phenomenological model, we can es￾timate NJJ in our inductor strips as shown in Fig. 4b from the measured self-Kerr coefficients. For all devices, w…
Figure 5
Figure 5. Figure 5: a as a function of input drive power for a critically coupled resonator. We find that the measured loss is power dependent and decreases monotonically with nph. At higher powers (nph > 100), the response becomes highly non-linear, and we do not approach a high-power Qi…
Figure 6
Figure 6. Figure 6: b, we show the single-photon quality factors of the grAl/Al and grAl/Ta hybrid resonators fabricated using this bandage technique together with the all-grAl devices. The highest quality factor we measure is 4.7 × 106 for a hybrid grAl/Ta sample at ρn ∼ 580 µΩcm. Intere…
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p015_14.png]

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