Graphene SQUIDs: Tunable Quantum Interference

• Graphene SQUIDs are superconducting quantum interference devices that use graphene weak links, offering tunable carrier densities and phase coherence via electrostatic gating.
• Their fabrication involves advanced assembly techniques including dry-pick-up of twisted bilayer/trilayer graphene and independent top- and backgate control for precise Josephson junction tuning.
• The devices exhibit high kinetic inductance, robust interference patterns, and potential applications in quantum electronics, high-field magnetometry, and topological superconductivity.

Graphene SQUIDs—superconducting quantum interference devices utilizing graphene as the weak link(s) within a Josephson interferometer—exploit the tunability, high carrier mobility, and correlation effects uniquely accessible in van der Waals heterostructures and moiré flat-band superconductors. Recent work demonstrates that twisted graphene systems, including twisted bilayer graphene (TBG), twisted trilayer graphene (TTG), and double-layer graphene (DLG), enable precise electrostatic control of device parameters, reaching kinetic inductances and phase coherence properties that far surpass those of conventional metallic or disordered superconductors. These advances position graphene-based SQUIDs as a versatile platform for phase-sensitive measurements, quantum electronics, and explorations of unconventional superconductivity.

1. Device Architectures and Fabrication

The architecture of graphene-based SQUIDs varies with the graphene system and targeted phenomenology, but all retain the essential SQUID loop with two graphene-based Josephson junctions (JJs). In TTG SQUIDs, the device comprises a MoRe (molybdenum–rhenium) superconducting loop incorporating two etched weak links of TTG, with each link independently tunable by local topgates and further controlled by a global graphite backgate. The TTG itself is realized by a dry-pick-up assembly of monolayer graphene sheets, with the central layer twisted by ≈1.7° to the adjacent layers, encapsulated between hBN, and contacted by Ti/Au topgates via an ALD-grown Al₂O₃ dielectric (Jha et al., 2024).

In monolithic MATBG SQUIDs, a magic-angle TBG stack is formed, with two independently gated arms defining the JJs of a ring-geometry SQUID. Top and bottom hBN encapsulation, Cr/Au edge contacts, and dual topgates allow full in situ electrostatic control of both junctions. Device dimensions include outer loop sizes of several microns and JJ segments ≈120 nm in length (Portolés et al., 2022).

Planar van der Waals SQUIDs employ monolayer and few-layer graphene as parallel weak links between exfoliated, mechanically split NbSe₂ superconducting contacts, all assembled on hBN and operated with a global SiO₂ back-gate. The fully planar geometry permits direct investigation of atomic-scale planarity effects and enables high-field operation due to the robust Ising spin–orbit coupling of NbSe₂ (Zalic et al., 2022).

Double-layer graphene SQUIDs utilize two parallel monolayer graphene sheets, separated by hBN and contacted by common MoRe superconducting leads, with loop areas defined by the layer separation and junction length. Both layers and their JJs are independently gated, enabling multichannel control and access to topological effects (Indolese et al., 2020).

2. Theory and Current–Phase Relation

The current–phase relation (CPR) describes the supercurrent $I(\theta)$ through a Josephson weak link as a function of the phase difference $\theta$. In proximity-induced regimes (short SNS-type junctions with low transparency), the CPR is sinusoidal: $I(\theta) = I_c \sin\theta$. In intrinsic superconducting links formed by TTG or MATBG at or near the magic angle, the CPR is strongly non-sinusoidal and approaches a piecewise-linear form, directly reflecting reduced superfluid density and strong kinetic inductance (Jha et al., 2024).

The device operation in the high asymmetry regime ($I_{c,2} \gg I_{c,1}$) allows direct mapping of the weak link's CPR from measured $I_c(B)$ versus magnetic flux. A universal relationship between the kinetic inductance $L_k$ and the critical current density $J_c$ is established:

$\lambda_k = \frac{\hbar}{3\sqrt{3}\ e\ \xi\ J_c}$

where $\lambda_k$ is the specific kinetic inductance, $\xi$ the coherence length, and $\theta$0 the critical current density, as derived within Ginzburg–Landau theory for thin films.

In DLG SQUIDs, the CPR is measurably skewed, indicating high-transparency Andreev modes with channel transparencies $\theta$1 depending on carrier type. The skewness and higher harmonic content in the CPR differentiate short, ballistic junctions from longer or more disordered ones. The CPR also serves as a sensitive probe of the symmetry of the underlying superconducting order and electronic transmission properties of the weak link (Indolese et al., 2020).

3. Gate-Tunability and Experimental Protocols

A hallmark of graphene SQUIDs is the ability to finely tune device properties with electrostatic gates. In TTG and MATBG SQUIDs, separate topgates on each weak link arm allow independent control of carrier density and displacement field, switching each arm between electron- and hole-like intrinsic superconductivity, or into a high-doping, proximity-induced regime. This results in a tunable superconducting dome in gate voltage space, with peak critical currents and kinetic inductance values reached near specific moiré fillings (e.g., $\theta$2) (Jha et al., 2024).

SQUID interference patterns, measured as oscillations in $\theta$3 vs. magnetic flux, directly reveal the effective charge of the superconducting carriers. In MATBG, the observed flux periodicity matches $\theta$4, consistent with Cooper pairs as superconducting charge carriers (Portolés et al., 2022). Gate tuning enables switching between symmetric and highly asymmetric SQUID regimes, altering the visibility and depth of the interference lobes.

The in situ tunability extends to inductance: in TTG SQUIDs, sheet kinetic inductance $\theta$5 is controlled over an order of magnitude (maximum values $\theta$6) by gating the junctions, enabling rapid switching between low and high impedance states (Jha et al., 2024).

4. Kinetic Inductance, Coherence, and Performance Metrics

The kinetic inductance $\theta$7 in graphene SQUIDs is exceptionally large, arising from both flat-band physics and suppressed superfluid density. In TTG, $\theta$8 achieves up to $\theta$9, exceeding values in TiN, NbN, or granular Al films (typically $I(\theta) = I_c \sin\theta$0) by orders of magnitude (Jha et al., 2024). MATBG SQUIDs similarly exhibit microhenry-scale kinetic inductance in micron-scale loops (Portolés et al., 2022). The tunability of $I(\theta) = I_c \sin\theta$1 is essentially tied to the gate-tuned $I(\theta) = I_c \sin\theta$2 through the universal $I(\theta) = I_c \sin\theta$3 relation, with extracted coherence lengths $I(\theta) = I_c \sin\theta$4 up to $I(\theta) = I_c \sin\theta$5, implying a uniform current flow and low phase-slip rates.

The unique high-field robustness of graphene-based SQUIDs is exemplified in planar graphene–NbSe₂ devices, which retain full interference up to in-plane fields $I(\theta) = I_c \sin\theta$6 exceeding $I(\theta) = I_c \sin\theta$7, due to the ISOC and atomic thickness of NbSe₂ (Zalic et al., 2022).

Current density distributions $I(\theta) = I_c \sin\theta$8 within the weak links can be reconstructed via maximum-entropy phase retrieval or autocorrelation analysis, revealing edge-enhanced or field-driven localization of supercurrent channels.

5. Interference Patterns, Quantum Hall Effect, and Topological Regimes

Graphene SQUIDs can operate both in the conventional Josephson regime, exhibiting standard SQUID and Fraunhofer oscillations, and in quantum Hall regimes supporting helical edge modes. In DLG-SQUIDs, perpendicular field induces quantized conductance plateaus at $I(\theta) = I_c \sin\theta$9 for specific filling factors, reflecting counter-propagating helical edge channels in spatially separated graphene layers (Indolese et al., 2020).

Under in-plane magnetic fields, the SQUID signal displays interference periodicity directly related to the area enclosed between the two graphene layers; in DLG-SQUIDs, this area matches $I_{c,2} \gg I_{c,1}$0 with $I_{c,2} \gg I_{c,1}$1 the junction length and $I_{c,2} \gg I_{c,1}$2 the interlayer separation.

Superconducting coherence and the CPR persist even in the quantum Hall regime, although supercurrent pockets disappear as phase coherence is lost near charge neutrality or at specific edge configurations, reflecting the underlying transmission of the edge states and proximity effect parameters.

6. Applications and Prospects for Quantum Technologies

Graphene SQUIDs offer unique features for quantum circuit engineering:

• High and gate-tunable kinetic inductance: Enables implementation of compact superinductors, field-effect-tunable qubit frequencies, and dynamically reconfigurable resonator impedances (Jha et al., 2024).
• Scalable quantum interference and sensing: Devices demonstrate micron-scale phase-coherent supercurrent routing and flux sensitivity, extensible to quantum bits and nanoscale magnetometers (Portolés et al., 2022).
• High magnetic field compatibility: The stability of planar graphene–NbSe₂ SQUIDs at $I_{c,2} \gg I_{c,1}$3 above $I_{c,2} \gg I_{c,1}$4 unlocks applications in high-field magnetometry and current-density imaging (Zalic et al., 2022).
• Topological superconductivity: DLG-SQUIDs in the quantum Hall regime provide an avenue for engineering topological superconductivity and probing Majorana modes via the coupling of helical edge states and Josephson contacts (Indolese et al., 2020).

Limitations include variability in twist-angle homogeneity, device reproducibility, and challenges in large-scale integration of atomically assembled van der Waals heterostructures. Future research directions involve expansion to other moiré superconductors, incorporation into high-frequency quantum circuits, and systematic exploration of unconventional superconducting order via phase-sensitive measurements and advanced CPR reconstructions.

7. Summary Table: Salient Characteristics of Graphene SQUID Platforms

Device Type	Max Kinetic Inductance	Key Tunable Elements
TTG SQUID (Jha et al., 2024)	$I_{c,2} \gg I_{c,1}$5	Carrier density, displacement field
MATBG SQUID (Portolés et al., 2022)	Several $I_{c,2} \gg I_{c,1}$6H (loop scale)	Junction inductance, critical current
DLG SQUID (Indolese et al., 2020)	$I_{c,2} \gg I_{c,1}$7A$I_{c,2} \gg I_{c,1}$8 at max	Layer densities, interlayer spacing
Graphene–NbSe₂ SQUID (Zalic et al., 2022)	High-field stable ($I_{c,2} \gg I_{c,1}$9 T)	Back-gate (density), field tuning

These characteristics highlight the diversity and technical potential of graphene-based SQUIDs as both scientific probes and building blocks for quantum technologies.