Granular Aluminum (grAl) Superconductor

• Granular aluminum (grAl) is a superconducting thin film composed of nanoscale Al grains separated by thin aluminum-oxide barriers that form an effective network of Josephson junctions.
• Its tunable normal-state resistivity and enhanced critical temperature enable precise control over superconducting properties and kinetic inductance, benefiting quantum devices like superinductors and resonators.
• Recent studies reveal grAl’s versatile electrodynamics, marked by strong nonlinearity, magnetic-field resilience, and complex phase coherence behavior near the superconductor–insulator transition.

Granular aluminum (grAl) is a superconducting thin film composed of nanometer-sized aluminum grains separated by thin aluminum-oxide barriers. Across the literature, it is described as a granular superconductor, a strongly disordered superconductor, and an effective self-assembled three-dimensional network of Josephson junctions. This mixed granular/disordered character gives grAl an unusual combination of properties: tunable normal-state resistivity, enhanced critical temperature over pure aluminum across part of its phase diagram, very large kinetic inductance, strong but often manageable nonlinearity, compatibility with conventional superconducting-circuit fabrication, and resilience in magnetic field. These features have made grAl a central material platform for superinductors, high-impedance resonators, protected-qubit circuits, parametric devices, kinetic-inductance detectors, and superconductor-semiconductor hybrids (Khorramshahi et al., 9 Jun 2025, Deshpande et al., 2024, Grünhaupt et al., 2018).

1. Material constitution, growth, and tunability

Granular aluminum is formed by depositing aluminum in an oxygen atmosphere so that the film self-assembles into nanoscale Al grains separated by thin AlO$_x$ barriers. Several studies describe grains of about $3$–$5$ nm, while other reports give grain sizes around $2$–$3$ nm or $6 \pm 1$ nm depending on growth and characterization method (Maleeva et al., 2018, Borisov et al., 2020, Bachar et al., 2014, Ramos et al., 5 Aug 2025). Physically, the material is best viewed as a random network of superconducting Al islands coupled through tunnel barriers, and on the microscopic level it resembles a Josephson-junction array, while on macroscopic length scales it behaves as a strongly disordered superconductor (Deshpande et al., 2024, Gupta et al., 2024).

Its central control parameter is disorder mediated by oxidation and growth conditions. Oxygen partial pressure and deposition rate tune the room-temperature sheet resistance or resistivity, and thereby tune the inter-grain coupling, kinetic inductance, critical temperature, and microwave response (Kamenov et al., 2019, Ramos et al., 5 Aug 2025). Lower substrate temperature during deposition provides an additional tuning knob: films grown on sapphire at $300\ \mathrm{K}$, $150\ \mathrm{K}$, $100\ \mathrm{K}$, and $25\ \mathrm{K}$ show that the maximum of the superconducting dome rises as the substrate temperature is reduced, reaching $3$0 for films grown at $3$1 (Deshpande et al., 2024).

The accessible resistivity range is exceptionally broad. In microwave-device work, grAl is often operated from tens to thousands of $3$2, for example $3$3 to $3$4 in cQED resonator studies, roughly $3$5 to $3$6 in high-impedance ring resonators, and approximately $3$7 to $3$8 in compact lumped-element inductors (Maleeva et al., 2018, Khorramshahi et al., 9 Jun 2025, Gupta et al., 2024). Other materials-focused work reports resistivities of order $3$9 with kinetic inductances of order $5$0, emphasizing that superconductivity can persist to very high resistivity in this granular system (Moshe et al., 2020).

This tunability underlies a recurrent practical distinction. In most circuit implementations, grAl is not the tunneling element in the conventional Al/AlO$5$1/Al sense; rather, it is the high-kinetic-inductance material that defines the circuit’s inductive or nonlinear environment (Kamenov et al., 2019, Grünhaupt et al., 2018). A separate line of work, however, uses a lithographically defined grAl constriction itself as a self-structured nano-junction, showing that grAl can also supply the Josephson nonlinearity directly in a fluxonium-like qubit (Rieger et al., 2022).

2. Phase diagram, superconducting dome, and the superconductor–insulator transition

A defining feature of grAl is the dome-shaped dependence of $5$2 on normal-state resistivity. As $5$3 increases, $5$4 first rises, reaches a maximum, and then falls again (Deshpande et al., 2024, Levy-Bertrand et al., 2019). Near the dome maximum, values of $5$5 around $5$6–$5$7 K are reported, exceeding the $5$8 of pure bulk aluminum, and colder growth raises the dome maximum further to $5$9 (Deshpande et al., 2024). The dome maximum remains in roughly the range $2$0 to $2$1 across the substrate temperatures studied in that work (Deshpande et al., 2024).

The standard interpretation is an interplay between the superconducting gap $2$2, the phase stiffness $2$3, and Coulomb charging effects. On the low-resistivity side, reduced grain size can enhance $2$4 and hence raise $2$5; on the high-resistivity side, stronger decoupling of grains suppresses superfluid density and superfluid stiffness, and $2$6 then falls (Deshpande et al., 2024). Optical and transport work formulates this competition with

$2$7

and an estimated charging energy

$2$8

with $2$9, $3$0 nm, and grain size $3$1–6 nm, giving roughly $3$2 (Levy-Bertrand et al., 2019).

Whether the superconductor-to-insulator transition is best regarded as disorder-driven or charging-driven remains a central issue. Several papers argue explicitly for a Mott-type or correlation-controlled transition in grAl, in contrast to the Anderson-type transition usually discussed for atomically disordered superconductors such as NbN$3$3 (Moshe et al., 2020, Bachar et al., 2014). The Mott-transition study reports that the effective Fermi energy inferred from magnetoresistance scales as

$3$4

and reaches the grain charging energy $3$5 near $3$6, where a metal-to-insulator transition is known to exist (Bachar et al., 2014). Scanning tunneling microscopy adds direct microscopic evidence: above the Mott resistivity $3$7, Coulomb charging effects and in-gap states appear on individual grains, which the authors interpret as a first indication of decoupling (Yang et al., 2019).

The local spectroscopy is notable because the superconducting gap remains robust on individual grains even in oxygen-rich samples. Pure Al shows $3$8, whereas grAl near $3$9 shows $6 \pm 1$0, and oxygen-rich grAl near $6 \pm 1$1 gives $6 \pm 1$2 (Yang et al., 2019). This supports a picture in which pairing within grains can remain strong while intergrain coherence progressively weakens.

Electrodynamic measurements near the SIT further show that grAl develops well-resolved sub-gap absorption features. In addition to the pair-breaking threshold at $6 \pm 1$3, two sub-gap modes are reported: $6 \pm 1$4, interpreted as likely related to two-dimensional plasma phase modes, and a broader mode $6 \pm 1$5 with $6 \pm 1$6, which softens with increasing resistivity and is associated with phase fluctuations (Levy-Bertrand et al., 2019). This suggests that the insulating approach is governed not simply by the disappearance of pairing, but by the collapse of phase coherence and the increasing importance of charging.

3. Electrodynamics, kinetic inductance, and nonlinear response

The large inductance of grAl is predominantly kinetic. In high-impedance resonators, the kinetic inductance fraction

$6 \pm 1$7

approaches unity (Khorramshahi et al., 9 Jun 2025). Equivalent conclusions appear in other microwave studies, with $6 \pm 1$8 for high-resistivity resonators near the SIT and $6 \pm 1$9 for compact lumped-element resonators (Grünhaupt et al., 2018, Gupta et al., 2024). In this regime, the circuit response is strongly sensitive to the superfluid density, quasiparticles, and current-induced nonlinearity.

Several related expressions are used for the kinetic inductance. One form written for grAl ring resonators is

$300\ \mathrm{K}$0

with $300\ \mathrm{K}$1, $300\ \mathrm{K}$2 for low resistivity, and $300\ \mathrm{K}$3 at higher resistivity (Khorramshahi et al., 9 Jun 2025). A closely related dirty-limit expression used in materials-oriented work is

$300\ \mathrm{K}$4

which makes explicit that larger $300\ \mathrm{K}$5 and smaller $300\ \mathrm{K}$6 raise the sheet kinetic inductance (Moshe et al., 2020). In compact circuit-QED inductors, the strip contribution is written as

$300\ \mathrm{K}$7

(Gupta et al., 2024)

Microwave spectroscopy confirms that the material behaves like an effective Josephson network. A one-dimensional effective-array description gives the mode dispersion

$300\ \mathrm{K}$8

with saturation at an effective plasma frequency

$300\ \mathrm{K}$9

In one representative stripline resonator, $150\ \mathrm{K}$0, $150\ \mathrm{K}$1, and $150\ \mathrm{K}$2 were extracted from the nonlinear dispersion (Maleeva et al., 2018).

The nonlinearity of grAl is strong but widely tunable. Early cQED measurements reported self-Kerr coefficients spanning $150\ \mathrm{K}$3 to $150\ \mathrm{K}$4, with a representative value $150\ \mathrm{K}$5 for one stripline resonator (Maleeva et al., 2018). Later compact inductor work found self-Kerr nonlinearities of $150\ \mathrm{K}$6–$150\ \mathrm{K}$7 for lumped-element resonators with sheet inductances $150\ \mathrm{K}$8–$150\ \mathrm{K}$9 pH/sq (Gupta et al., 2024). High-impedance ring resonators then pushed to impedances above $100\ \mathrm{K}$0 while keeping the self-Kerr coefficient $100\ \mathrm{K}$1 in the range of tens of Hz (Khorramshahi et al., 9 Jun 2025). In that work, higher resistivity and thinner films give larger $100\ \mathrm{K}$2, consistent with

$100\ \mathrm{K}$3

The kinetic inductance is also explicitly current dependent. In a $100\ \mathrm{K}$4 mm grAl coplanar transmission line at $100\ \mathrm{K}$5, the measured microwave phase shift reached $100\ \mathrm{K}$6 radians at a bias current of about $100\ \mathrm{K}$7, corresponding to a relative nonlinearity of about $100\ \mathrm{K}$8 (Zhdanova et al., 2024). The current dependence was modeled both with the low-current expansion

$100\ \mathrm{K}$9

with fitted $25\ \mathrm{K}$0, and with a dirty-limit BCS-based form involving the depairing current, with $25\ \mathrm{K}$1 (Zhdanova et al., 2024). This nonlinear kinetic-inductance response underlies grAl parametric amplifiers and other strongly driven microwave devices.

4. High-impedance resonators, superinductors, and compact inductive elements

A principal technological role of grAl is as a superinductor material. Superinductors are inductors with impedance exceeding the resistance quantum

$25\ \mathrm{K}$2

or $25\ \mathrm{K}$3–$25\ \mathrm{K}$4 depending on the paper’s quoted precision (Khorramshahi et al., 9 Jun 2025, Kamenov et al., 2019, Grünhaupt et al., 2018). grAl achieves this by combining high sheet inductance with low parasitic capacitance in compact geometries.

An early demonstration used meandered nanowires for hybrid quantum circuits. The reported headline performance was total inductance $25\ \mathrm{K}$5 with self-resonance above $25\ \mathrm{K}$6; measured devices reached $25\ \mathrm{K}$7 and $25\ \mathrm{K}$8 with $25\ \mathrm{K}$9 and $3$00, corresponding to impedances $3$01 and $3$02 (Kamenov et al., 2019). The devices occupied footprints not exceeding $3$03, showing that large inductance could be realized in a very small on-chip area (Kamenov et al., 2019).

The 2025 ring-resonator work substantially extended this regime. Meandered grAl traces patterned at pitches of $3$04–$3$05 nm, widths of $3$06–$3$07 nm, and thicknesses of $3$08–$3$09 nm produced extracted sheet inductances from $3$10 to $3$11, total inductance as high as $3$12, and a largest measured impedance of $3$13 in the $3$14–$3$15 band (Khorramshahi et al., 9 Jun 2025). The impedance was extracted using

$3$16

For these devices the impedance exceeds $3$17 by about a factor of $3$18 (Khorramshahi et al., 9 Jun 2025).

That work also states a simple geometric scaling model: $3$19 The design logic is explicit: reduce inner radius $3$20, narrow the trace $3$21, reduce pitch $3$22, and increase sheet inductance $3$23 to raise the impedance, while accepting an upward shift in resonance frequency (Khorramshahi et al., 9 Jun 2025). The same paper outlines that an optimized planar grAl resonator on Si or sapphire could reach about $3$24 in the $3$25–$3$26 band, and that freestanding or membrane-supported release could increase the impedance by about a factor of three, exceeding $3$27 (Khorramshahi et al., 9 Jun 2025).

A complementary device direction uses grAl for compact but lower-impedance lumped inductors in conventional circuit QED. Lumped-element resonators entirely made from grAl, or hybridized with Al or Ta capacitor electrodes, were reported with $3$28–$3$29, $3$30–$3$31, $3$32 often around $3$33–$3$34, and impedance in the $3$35–$3$36 range (Gupta et al., 2024). The smallest photolithographic grAl inductors were only $3$37 wide and $3$38 long, achieved several nH of inductance, and were up to $3$39 more compact than comparable inductors made from pure Al geometric inductance (Gupta et al., 2024). This establishes grAl not only as a route to superinductance, but also as a route to linear nH-scale inductors in micrometer-scale footprints.

5. Circuit implementations: qubits, parametric devices, and hybrid semiconductor structures

grAl has been integrated into several classes of superconducting quantum circuits. In a fluxonium implementation, a $3$40-long grAl strip served as the superinductor shunting a small Al/AlO$3$41/Al junction (Grünhaupt et al., 2018). Spectroscopy gave $3$42, $3$43, and $3$44, with qubit transition frequency tunable from $3$45 at integer flux to $3$46 at half-integer flux (Grünhaupt et al., 2018). The first self-resonant mode of the strip occurred at $3$47, sufficiently above the qubit band to treat the superinductor as a lumped element (Grünhaupt et al., 2018). Coherence times reached $3$48, $3$49 in the main text and up to $3$50 in the summary, and $3$51 (Grünhaupt et al., 2018).

A more radical use of grAl replaces the conventional Josephson junction itself. The “gralmonium” device uses a single-layer, zero-angle-deposited grAl nano-junction in a fluxonium architecture with Hamiltonian

$3$52

For the main device, the reported parameters are $3$53, $3$54, $3$55, and $3$56 (Rieger et al., 2022). The measured spectrum is described as indistinguishable from that of a standard fluxonium qubit, higher harmonics in the current-phase relation contribute less than about $3$57, and coherence values are $3$58 scale with $3$59 around $3$60–$3$61 (Rieger et al., 2022). The same work reports spontaneous jumps in $3$62 on timescales from milliseconds to days, which offer a direct probe of microscopic defects (Rieger et al., 2022).

grAl can also supply the nonlinearity of a transmon-like qubit. A small grAl volume $3$63, shunted by a thin-film Al capacitor, produced a microwave oscillator with fundamental transition $3$64, anharmonicity $3$65, linewidth $3$66, and intrinsic linewidth $3$67 (Winkel et al., 2019). Time-domain measurements gave $3$68, $3$69, and $3$70, while fluorescence measurements implied an intrinsic lifetime of about $3$71 (Winkel et al., 2019). The intrinsic linewidth remained below $3$72 for in-plane magnetic fields up to $3$73 (Winkel et al., 2019).

In parametric amplification, grAl provides kinetic-inductance nonlinearity without relying on field-fragile tunnel-junction critical currents. A non-degenerate amplifier built from two coupled grAl resonators demonstrated $3$74 gain close to the quantum limit of added noise, a gain-bandwidth product of $3$75, and $3$76 input saturation power, while remaining resilient to in-plane magnetic fields up to $3$77 (Zapata et al., 2024). The local Kerr coefficients remain in the $3$78–$3$79 range up to $3$80, and the hopping coupling $3$81 changes by less than $3$82 over the full field range (Zapata et al., 2024). This operational regime is consistent with the independent observation of current-dependent kinetic-inductance nonlinearity in long grAl transmission lines (Zhdanova et al., 2024).

grAl has also become a superconducting parent material in semiconductor hybrid structures. Depositing grAl on Ge/SiGe heterostructures induces a hard superconducting gap in planar Ge, with BCS peaks at $3$83, subgap conductance about two orders of magnitude lower than the out-of-gap conductance, and subgap conductance below the noise floor $3$84 at zero bias under the best field conditions (Fabris et al., 24 Feb 2026). The induced gap $3$85 is about $3$86 of the grAl film gap $3$87 (Fabris et al., 24 Feb 2026). Magnetic-field resilience is strong in both orientations: the proximitized Ge contact kept zero-bias conductance below noise up to $3$88 out-of-plane and $3$89 in-plane for a $3$90 wide, $3$91 thick grAl contact (Fabris et al., 24 Feb 2026). In gate-defined Ge quantum dots, this enabled Yu–Shiba–Rusinov physics, Zeeman splitting of YSR states beyond $3$92 ($3$93), and extraction of hole $3$94-factors $3$95 and $3$96, with maximum in-plane $3$97-factor up to $3$98 (Fabris et al., 24 Feb 2026).

6. Dissipation, quasiparticles, TLS, magnetic-field response, and unresolved issues

Despite the material’s high inductance and broad utility, dissipation and noise remain major themes in grAl research. High-kinetic-inductance resonators close to the SIT show single-photon internal quality factors on the order of $3$99, but time-domain monitoring revealed sudden downward jumps in resonance frequency roughly every $5$00, followed by slow recovery over seconds (Grünhaupt et al., 2018). The interpretation is that non-equilibrium quasiparticles dominate the microwave loss. The work models mobile and localized quasiparticle populations with coupled rate equations,

$5$01

$5$02

and extracts relaxation times roughly $5$03 at $5$04 and about $5$05 to $5$06 in the single-photon regime (Grünhaupt et al., 2018). The average burst interval is about $5$07 (Grünhaupt et al., 2018).

Phonon engineering can mitigate this. Surrounding grAl resonators with lower-gap Al islands, placed far enough away to remain electromagnetically decoupled, increases the single-photon $5$08, suppresses low-frequency noise, and reduces the rate of quasiparticle bursts (Henriques et al., 2019). The paper reports more than factor-of-two improvement in single-photon $5$09, summarizes improvements up to a factor of three, observes about an order-of-magnitude reduction in low-frequency noise amplitude, and finds the burst rate roughly halved for $5$10 trap coverage, while the quasiparticle lifetime remains $5$11 for all $5$12 (Henriques et al., 2019). The interpretation is that the traps reduce quasiparticle generation by downconverting substrate phonons rather than changing recombination dynamics.

Other regimes show a different loss phenomenology. In low-loss lumped-element inductors, the best all-grAl resonators reached $5$13 in the summary and $5$14 in the table, while hybrid grAl/Ta resonators reached $5$15 in the summary and $5$16 as a measured maximum (Gupta et al., 2024). However, the internal quality factor systematically decreases with room-temperature resistivity, from about $5$17 at $5$18 down to about $5$19 at $5$20 (Gupta et al., 2024). The loss model

$5$21

supports a picture in which low-resistivity films have surface loss factors similar to pure Al, while increasing loss at larger resistivity can be explained by increasing conductor loss in the grAl strip (Gupta et al., 2024). This constitutes a concrete compactness-versus-loss trade-off.

TLS-related loss is also clearly observed. Coplanar resonators fabricated from grAl with different oxygen content show power-dependent internal quality factors consistent with TLS saturation,

$5$22

with average TLS participation factor $5$23, $5$24, and saturation below roughly one photon (Ramos et al., 5 Aug 2025). That work explicitly attributes the TLS to impurities, defects, dangling bonds, and interfaces inherent to the oxygen-assisted growth process (Ramos et al., 5 Aug 2025). By contrast, the high-resistivity quasiparticle work argued that dielectric loss was not the dominant mechanism there, because large changes in metal-substrate participation ratio had little effect until the participation was increased by about two orders of magnitude (Grünhaupt et al., 2018). Taken together, these studies show that the dominant loss mechanism depends strongly on geometry, resistivity, and operating regime.

Magnetic-field response is another area where grAl departs from conventional Al. $5$25 grAl resonators with film thickness $5$26, kinetic inductance $5$27–$5$28, and $5$29 retain single-photon $5$30 in in-plane fields up to $5$31 in one waveguide and $5$32 in another (Borisov et al., 2020). Small perpendicular fields around $5$33–$5$34 enhance $5$35 by about $5$36, possibly because fluxons act as quasiparticle traps, whereas larger perpendicular fields induce flux trapping, long-time drifts, and random $5$37 changes (Borisov et al., 2020). An electron-spin-resonance dip with extracted $5$38 was attributed to a spin-$5$39 ensemble, possibly spins localized in the oxide between Al grains (Borisov et al., 2020). This is consistent with the broader microscopic literature reporting free spins by $5$40SR and YSR-like in-gap states in oxygen-rich grAl (Bachar et al., 2014, Yang et al., 2019).

Several open issues remain unresolved. In some high-impedance ring resonators, a positive Kerr coefficient and an anomalous positive frequency shift at very low photon number were observed in some devices and cooldowns, with no clear correlation to impedance, resistivity, or geometry; the origin is unknown, though strongly coupled spurious two-level systems were suggested as a possibility (Khorramshahi et al., 9 Jun 2025). STM studies report multiple low-energy states outside the gap that may indicate bosonic excitations of the superconducting order parameter, but no firm conclusion is drawn (Yang et al., 2019). In insulating grAl, low-temperature conductance shows glassy nonequilibrium dynamics with thermal activation and activation energies up to about $5$41, calling into question purely quantum-glass interpretations for this class of disordered insulators (Grenet et al., 2017). These unresolved aspects are significant because the same microscopic ingredients—charging, localized spins, sub-gap states, and slow configurational dynamics—are plausible contributors to flux noise, dielectric loss, and low-power instabilities in quantum circuits.

A plausible implication is that grAl’s central scientific interest and its technological value are inseparable. The same nanogranular structure that yields exceptionally large kinetic inductance, high impedance, and magnetic-field resilience also creates a parameter space in which phase stiffness, charging, quasiparticles, localized spins, and sub-gap excitations all become experimentally relevant.