Moiré Superconductivity

Moiré superconductivity is phase-coherent charge-$2e$ condensation in an electronic structure for which a moiré superlattice materially controls the active bands, interactions, or order parameter. The definition excludes two common overextensions: a resistance anomaly without demonstrated coherence, and ordinary superconductivity in a layer whose moiré modulation is incidental.

Two questions guide the discussion. What binds the electrons into pairs, and what makes the pairs move coherently across the device? A narrow band can strengthen pairing while providing little conventional phase stiffness. Moiré devices make this distinction experimentally useful because gates tune filling, layer character and weak links in the same sample.

Required background. Moiré Superlattices supplies the miniband, density and gate-controlled energy scales. Flat Bands supplies the distinction between narrow dispersion, Bloch-state geometry and interaction effects. BCS Theory supplies the paired-state and gap baseline against which superconducting material observations are compared.

Establishing a Superconducting Phase

A coherent current through the device

A four-terminal resistance that falls below the voltage resolution over a finite current range is stronger evidence than a partial downturn. The measurement should be repeated across excitation currents, contact pairs, thermal cycles, and magnetic fields. Current bias must remain below the switching scale; otherwise an apparent residual resistance can be self-induced.

Nonlinear differential resistance can reveal a low-bias zero-voltage branch and a switching current. In an ultrathin, high-kinetic-inductance device, however, Joule heating and slow electronic cooling can mimic abrupt switching or hysteresis. A thermal model, pulse measurements, or frequency dependence is needed before interpreting every switching feature as a fundamental critical current.

Perpendicular magnetic field should introduce orbital depairing and vortices. A smoothly vanishing zero-resistance region, reproducible critical-field scale, vortex-flow response, or Josephson interference ties the transport anomaly to superconductivity. The convenient Ginzburg–Landau estimate $B_{c2}^{\perp}=\Phi_0/(2\pi\xi_{\mathrm{GL}}^2)$ defines an effective coherence length only near the regime where that theory applies; it is not a microscopic pair-size measurement.

Separate pairing, stiffness, and zero resistance

In a two-dimensional moiré band, the temperature at which a pairing amplitude becomes appreciable need not equal the temperature of global phase coherence. Use the Ginzburg–Landau convention: $\theta$ is the phase of the electron-pair field, with charge $q^\ast=-2e$ and $e>0$. Write its slowly varying phase energy as

$$\begin{aligned} F_\theta &= \frac12 \int d^2r\, \rho_{s,ij}q_iq_j, \\ q_i &= \partial_i\theta - \frac{q^\ast}{\hbar}A_i =\partial_i\theta+\frac{2e}{\hbar}A_i. \end{aligned}$$

The two-dimensional stiffness $\rho_s$ has units of energy: the area integral cancels the two inverse lengths from the phase gradients. The renormalized isotropic stiffness obeys the ideal Nelson–Kosterlitz (1977) jump $\rho_s(T_{\mathrm{BKT}}^-)=2k_{\mathrm B}T_{\mathrm{BKT}}/\pi$. This ideal BKT description assumes logarithmic vortex interactions over the length scales of interest. In a thin charged film, magnetic screening cuts that interaction off near the Pearl (1964) length $\Lambda=2L_k^\square/\mu_0$; finite device size can impose an earlier cutoff. Inhomogeneity, vortex pinning and heating also round the transition. For a uniform positive anisotropic stiffness tensor, rescaling the two coordinates gives the effective stiffness $\sqrt{\det\rho_s}$, so the ideal jump concerns that quantity, not either principal component separately. A current–voltage power law $V\propto I^{a(T)}$ with $a=3$ at an inferred transition is suggestive only when the fitted current window and thermal stability are reported.

Microwave response can determine the sheet kinetic inductance after separating the condensate response from normal carriers, dielectric losses and the rest of the circuit. With the phase-stiffness convention above,

$$L_k^\square = \frac{\hbar^2} {4e^2\rho_s}.$$

The scalar relation is for isotropic response; along a principal axis use its stiffness component. A finite-frequency inductance probes a finite time and length scale, so it need not equal the thermodynamic stiffness after all long-distance vortex fluctuations.

Tanaka et al. (2025) and Banerjee et al. (2025) use kinetic-inductance measurements in twisted bilayer and trilayer graphene, respectively. A direct stiffness measurement closes an important gap between transport and thermodynamics: it establishes phase rigidity and tests whether the observed transition is limited by pair breaking or by phase fluctuations.

Spectroscopy measures complementary gaps

Tunneling spectroscopy probes the one-particle density of states, while Andreev reflection probes electron-to-hole conversion at a superconducting interface. Their inferred gap scales need not coincide in a multiband, inhomogeneous, or pseudogapped system. A depletion that survives above the zero-resistance temperature can indicate preformed pairs, a competing order, or a normal-state pseudogap; temperature and magnetic-field evolution are required to distinguish them.

The ratio

$$R_\Delta = \frac{2\Delta} {k_{\mathrm B}T_c}$$

equals approximately $3.53$ for the weak-coupling isotropic gap at zero temperature in Bardeen, Cooper, and Schrieffer (1957). A larger value signals departure from that narrow benchmark, but it does not uniquely imply electronic pairing: strong electron–phonon coupling, anisotropy, multiple gaps, pseudogap contamination, and an underestimated phase-ordering temperature can all enlarge it.

Pairing and phase coherence need not disappear together. For the illustrative below-transition stiffness $\rho_s(T)=\rho_0(1-T/T_*)$, the ideal jump occurs where it meets $2k_BT/\pi$. Here $\rho_0=0.10\,\mathrm{meV}$ and $T_*=4\,\mathrm K$ give $T_{\mathrm{BKT}}=1.25\,\mathrm K$. The thin dashed continuation is an extrapolation, not a finite thermodynamic stiffness above the transition. This is a teaching model, not a fit to a material or a vortex-renormalization calculation.

Superconductivity Near Correlated Phases

Adjacency is a clue, not a mechanism

Cao et al. (2018) found superconducting domes near interaction-driven insulating states. That resemblance to cuprate phase diagrams made doped-Mott and fluctuation-mediated pictures natural. Later devices showed superconductivity when the nearby correlated insulator was weak or absent, as shown away from the flat-band condition by Saito et al. (2020) and in WSe$_2$-proximitized structures by Arora et al. (2020). Conversely, strong correlated states need not produce superconductivity.

A neighboring insulating phase suggests a mechanism to test, but both phases might respond to a third factor such as a van Hove singularity or a change in screening. Compare their evolution under tuning in the same device region and ask whether the same band and interaction parameters account for both. An insulating gap that disappears while superconductivity remains weakens the hypothesis that this particular ordered insulator is a necessary parent.

The same-region thermodynamic and transport measurements of Hoke et al. (2026) in twisted trilayer graphene strengthen such comparisons while also showing that correlated-insulator gaps and superconducting strength need not track one another. In twisted WSe$_2$, Xia et al. (2026) report superconducting domes adjoining a bandwidth-tuned antiferromagnetic insulator. These provide a tunable correlation, but they still do not by themselves identify the pairing boson or order-parameter symmetry.

The normal state selects available pairing channels

The superconducting phase emerges from a normal state that can contain flavor polarization, strange-metal transport, nematicity, pseudogaps, van Hove singularities, or multiple Fermi pockets. Those features alter the available Cooper channels.

A pairing theory should reproduce the normal-state Fermi surface and flavor occupancy at the same gate coordinates. Comparing a superconducting dome with a noninteracting band calculated at neutrality, while ignoring interaction-driven flavor reconstruction, can assign the wrong density of states, screening, and gap symmetry.

Tunability by Doping and Displacement Field

Dual gates define coupled coordinates

For top and bottom gate capacitances per area $C_t$ and $C_b$, a common electrostatic convention is

$$\begin{aligned} n &= \frac{ C_t(V_t-V_t^0) + C_b(V_b-V_b^0) }{e}, \\ D &= \frac12 \left[ C_t(V_t-V_t^0) - C_b(V_b-V_b^0) \right]. \end{aligned}$$

Here $D$ is an electric displacement with units of charge per area; many papers quote $D/\epsilon_0$ as an equivalent field. Sign conventions vary. Quantum capacitance, trapped charge, screening by nearby bands, and contact-gate geometry can modify this parallel-plate conversion.

Changing $n$ moves the chemical potential and can change flavor occupancy. Changing $D$ redistributes charge among layers, modifies hybridization and Berry curvature, and can shift van Hove singularities. In real dual-gated devices, the two coordinates are only approximately independent because compressibility and contact regions respond to both.

Twist, pressure, and screening are not scalar bandwidth knobs

Twist angle changes moiré period, tunneling interference, relaxation, wave-function geometry, and inhomogeneity. Hydrostatic pressure changes interlayer tunneling and structural relaxation. A nearby metallic gate shortens the long-range Coulomb tail and may alter the relative importance of repulsion, phonons, and disorder. Proximity to WSe$_2$ can introduce spin–orbit coupling while also changing dielectric screening and strain.

For that reason a measured $T_c$ should be written schematically as

$$T_c = T_c \!\left( \nu,D,\theta, \epsilon_{\mathrm{env}}, p,\varepsilon_{ij}, \Gamma,\ldots \right),$$

where $\Gamma$ represents disorder or broadening. Collapsing this parameter space into one inferred $U/W$ can hide the actual control variable.

Domes can have different boundaries

A superconducting dome may terminate at a flavor transition, a Lifshitz transition, a competing insulator, a loss of band isolation, a phase-stiffness threshold, or a disorder-dominated region. The same visual shape can therefore arise from different physics. Boundary tracking should combine Hall density, compressibility, spectroscopy, and symmetry probes rather than rely on resistance alone.

Pairing Structure

Fermionic antisymmetry is the first constraint

Assume a spatially uniform, zero-center-of-mass, even-frequency pair amplitude. Let $a,b$ collect spin, valley, layer, orbital and band labels. The pairing matrix

$$\Delta_{ab}(\mathbf k) = -\Delta_{ba}(-\mathbf k)$$

must be antisymmetric under exchange of the two electrons. More generally, $\Delta_{ab}(\mathbf k,i\omega_n)=-\Delta_{ba}(-\mathbf k,-i\omega_n)$; odd-frequency pairs require keeping that frequency sign. Even-parity momentum structure therefore pairs with an antisymmetric internal state, and odd parity with a symmetric one, after all active labels are included. Calling a state “spin singlet” does not fix its valley, layer, or orbital structure.

Momentum parity here describes exchange of relative momentum. It is a crystal-symmetry label only when the device has the corresponding inversion symmetry. The superconducting representation must be classified under the actual device symmetry. Heterostrain, substrate alignment, displacement field, and spontaneous normal-state order can reduce the ideal moiré point group. An apparently two-component order parameter in the ideal lattice may split into nondegenerate components in the fabricated device.

The linearized gap problem identifies leading channels

After specifying a normal-state Green function and irreducible pairing interaction, the transition can be organized as an eigenvalue problem:

$$\lambda(T)\Delta_\alpha(\mathbf k)=-\sum_{\beta,\mathbf k'}K_{\alpha\beta}(\mathbf k,\mathbf k';T)\Delta_\beta(\mathbf k').$$

The leading eigenvalue reaches one at a mean-field instability. The compound index $\alpha$ includes flavor and band structure. This calculation is predictive only if the kernel, screening, cutoff, normal-state reconstruction, and self-energy are controlled. Choosing a desired form factor and then fitting one coupling does not establish a mechanism.

Nodes, signs, and anisotropy are distinct

A nodeless excitation spectrum need not be conventional $s$ wave: sign changes can occur between disconnected pockets or flavors. Conversely, power-law temperature dependence can come from nodes, deep gap minima, pair breaking, or an inhomogeneous distribution of local gaps. Phase-sensitive junctions, quasiparticle interference, impurity response, and directional spectroscopy are needed to resolve the sign structure.

Nematic superconductivity breaks rotational symmetry in the condensate. Transport anisotropy that already exists above $T_c$ can instead reflect strain or normal-state nematicity inherited by an otherwise symmetry-preserving gap. The decisive test compares the symmetry onset and orientation across the transition and under controlled strain.

Pairing Mechanisms Under Debate

Phonon-mediated attraction

Graphene and TMD layers supply acoustic, optical, breathing, shear, and moiré-scale lattice modes. A large flat-band density of states can amplify electron–phonon pairing, while Coulomb repulsion, screening, vertex corrections and band geometry determine whether a weak-coupling treatment is controlled. A narrow electronic energy scale can invalidate the adiabatic separation behind the usual Migdal approximation; a large density of states alone does not justify neglecting vertex corrections.

For a concrete theoretical proposal, Wu, MacDonald, and Martin (2018) find phonon attraction in both $s$- and $d$-wave channels, with Coulomb terms affecting which wins. This is a model result, not a measured identification of the pairing interaction.

Isotope shifts, mode-resolved tunneling structures, phonon linewidth anomalies, and quantitative agreement among the measured coupling, gap, and $T_c$ would strengthen a phonon assignment. The observation of strong electron–phonon coupling is important but not sufficient: a phonon can dress quasiparticles without being the dominant pairing glue, and electronic interactions can reshape the same spectra.

Electronic fluctuations and repulsive pairing

Repulsive interactions can generate sign-changing attraction in selected angular, spin, or valley channels through spin fluctuations, valley fluctuations, charge fluctuations, exchange, or Kohn–Luttinger processes. Proximity to flavor-polarized or antiferromagnetic phases makes these channels plausible.

Isobe, Yuan, and Fu (2018) provide a distinct example: a hot-spot renormalization-group treatment near a van Hove regime produces competing superconducting and density-wave instabilities from repulsive interactions. The model assumptions matter when comparing it with a device.

Their strongest tests are correlated evolution of the relevant collective mode and $T_c$, a gap sign and symmetry matching the calculated leading eigenfunction, and quantitative scales from one normal-state interaction model. A dome near an insulator or a linear-in-$T$ metal is not itself evidence for a specific fluctuation.

Strong-coupling descendants of local models

When charge motion is strongly constrained near an integer filling, doping an extended Hubbard or exchange model can produce paired states. This language is especially natural for superconductivity emerging near a bandwidth-controlled Mott transition. Yet moiré orbitals are extended, interactions are nonlocal, and topological bands may obstruct a simple localized basis. “Doped Mott” should name a demonstrated model regime, not merely the shape of a phase diagram.

Quantum geometry and flat-band mobility

Flat dispersion enhances interaction effects but suppresses conventional band velocity. Within a specified multiband mean-field model, the phase-stiffness tensor can be organized schematically as

$$\rho_{s,ij}=\rho_{ij}^{\mathrm{conv}}+\rho_{ij}^{\mathrm{geom}}.$$

The conventional term follows dispersion; the geometric term depends on interband matrix elements and the quantum metric of the active projectors. The multiband mean-field result of Peotta and Törmä (2015) shows how quantum geometry can support phase stiffness even in a flat band under the model’s pairing assumptions. It is not, by itself, an attractive interaction and does not choose the spin, valley, or momentum symmetry of the pair.

Electronic repulsion can renormalize a phonon-mediated channel, phonons can select among nearly degenerate pair states, and quantum geometry can provide stiffness after pairs form. The useful distinction is between a mechanism that supplies attraction and the wavefunction structure that lets a paired state carry current.

Josephson and Device Probes

Gate-defined weak links

de Vries et al. (2021) demonstrate gate-defined Josephson junctions within one magic-angle graphene crystal, allowing the superconducting leads and weak link to be patterned electrostatically. A local gate can tune the barrier through metallic, correlated-insulating, or flavor-polarized regimes. This avoids a conventional materials interface but does not remove interfaces altogether: density gradients, fringe fields, twist disorder, and contact-induced doping define an electrostatic junction.

DC supercurrent, microwave-induced Shapiro steps, and magnetic interference demonstrate phase-coherent coupling. The phase and circuit equations are developed in Josephson Effect. In a moiré junction, the additional scientific opportunity is to vary the weak-link order while keeping the surrounding crystal fixed.

SQUIDs and current–phase relations

Portolés et al. (2022) demonstrate a monolithic twisted-bilayer-graphene SQUID with two gate-defined junctions. Flux-periodic critical current demonstrates phase coherence around the loop and can test the effective transported charge. Tunable junction asymmetry can expose an individual current–phase relation.

Non-sinusoidal or phase-shifted behavior can arise from high transparency, multiple paths, magnetic weak links, spin–orbit coupling, trapped vortices, or unconventional pairing. A superconducting diode effect likewise requires broken inversion and time-reversal symmetry in the effective transport problem, but it does not uniquely identify where those symmetries are broken. Nonuniform current density and large kinetic inductance can imitate intrinsic phase shifts.

Resolving the sign of the order parameter

The decisive pairing-symmetry experiments compare junctions with controlled crystallographic orientation, valley filtering, or known reference superconductors. A corner geometry, half-flux shift, or reproducible anomalous phase can test sign changes only after self-field, faceting, trapped flux, and magnetic weak-link alternatives are bounded. The extreme electrostatic tunability of moiré devices makes such controls possible, but also makes them sensitive to microscopic inhomogeneity.

Relation to Unconventional Superconductivity

Use an operational definition

“Unconventional” must name an axis. A nontrivial pairing-structure claim concerns crystal representation, parity or pseudospin content, nodes, relative signs, or multicomponent symmetry breaking. A pairing-mechanism claim concerns the interaction that generated the state and requires separate evidence. Strong coupling, a nonphononic proposal, or an unusual structure does not establish the other axis by itself.

A large gap ratio, a pseudogap, a nematic response or a large in-plane critical field each constrains a different assumption. For example, the weak-coupling Pauli estimate $B_P[\mathrm T]\simeq1.84\,T_c[\mathrm K]$ assumes a uniform weak-coupling spin-singlet state, $g=2$ and negligible spin–orbit coupling. Exceeding it invalidates that package of assumptions; it does not rule out spin-singlet pairing on its own. Unconventional Superconductivity explains how symmetry, gap sign and pairing mechanism are tested separately.

Small Fermi energy invites crossover physics

In a narrow band, $\Delta$ and $k_{\mathrm B}T_c$ can become non-negligible compared with an inferred Fermi energy. A small $k_F\xi$ and separated pairing and stiffness scales may suggest BCS–BEC crossover physics. Yet $E_F$ is ambiguous near van Hove points, interaction-reconstructed bands, and low-density pockets. A convincing crossover claim combines chemical potential, pair size, stiffness, spectral evolution, and fluctuation thermodynamics rather than quoting one ratio.

Complementary measurements

Measurement	Interpretation and main limitation
Four-terminal zero resistance	A dissipationless channel at the resolution used; it does not determine homogeneous superconducting volume.
Critical current	A collective current scale; heating can alter switching.
Calibrated Meissner response or condensate stiffness	Superconducting response on the measured scales, after normal backgrounds are removed; it does not identify the pairing interaction.
Tunneling or Andreev spectrum	Excitation or pair-conversion scales; a gap can persist without global coherence.
Josephson interference	Phase-coherent coupling; junction geometry and magnetic effects also shape the pattern.

The open pairing questions and dated developments are discussed in Moiré Quantum Systems.

Exercises

Exercise 1: dual-gate coordinates

Take $C_t=C_b=0.10\,\mu\mathrm F\,\mathrm{cm}^{-2}$, $V_t-V_t^0=1.5\,\mathrm V$, and $V_b-V_b^0=0.5\,\mathrm V$. Find $n$ and $D/\epsilon_0$. Use $e=1.602\times10^{-19}\,\mathrm C$ and $\epsilon_0=8.854\times10^{-12}\,\mathrm{F\,m^{-1}}$.

Solution

$0.10\,\mu\mathrm F\,\mathrm{cm}^{-2}=1.0\times10^{-3}\,\mathrm{F\,m^{-2}}$. Therefore

$$\begin{aligned} n &= \frac{ (1.0\times10^{-3})(2.0) }{1.602\times10^{-19}} \\ &= 1.25\times10^{16}\,\mathrm{m}^{-2}, \\ &= 1.25\times10^{12}\,\mathrm{cm}^{-2}. \end{aligned}$$

The displacement is

$$\begin{aligned} D &= \frac12 (1.0\times10^{-3})(1.0) \\ &= 5.0\times10^{-4}\,\mathrm{C\,m^{-2}}. \end{aligned}$$

so $D/\epsilon_0=5.65\times10^7\,\mathrm{V\,m^{-1}}=0.0565\,\mathrm{V\,nm^{-1}}$. A real-device analysis must also apply its sign convention and electrostatic corrections.

Exercise 2: BKT stiffness scale

Assuming the ideal isotropic jump, estimate $\rho_s(T_{\mathrm{BKT}}^-)$ for $T_{\mathrm{BKT}}=1.8\,\mathrm K$. Use $k_{\mathrm B}=0.08617\,\mathrm{meV\,K^{-1}}$.

Solution

The jump condition gives

$$\begin{aligned} \rho_s(T_{\mathrm{BKT}}^-) &= \frac{2}{\pi} k_{\mathrm B}T_{\mathrm{BKT}} \\ &= \frac{2}{\pi} (0.08617)(1.8)\,\mathrm{meV} \\ &\simeq 0.0987\,\mathrm{meV}. \end{aligned}$$

This is a renormalized long-wavelength stiffness immediately below the transition, not the zero-temperature bare-band value.

Exercise 3: sheet kinetic inductance

Using $\rho_s=0.10\,\mathrm{meV}$, estimate $L_k^\square$. Take $\hbar=1.055\times10^{-34}\,\mathrm{J\,s}$.

Solution

Because $0.10\,\mathrm{meV}=1.0\times10^{-4}\,\mathrm{eV}$ and $1\,\mathrm{eV}=1.602\times10^{-19}\,\mathrm J$, the stiffness is

$$\rho_s = 1.602\times10^{-23}\,\mathrm J.$$

Then

$$\begin{aligned} L_k^\square &= \frac{\hbar^2} {4e^2\rho_s} \\ &\simeq 6.8\times10^{-9}\,\mathrm H. \end{aligned}$$

Thus the sheet inductance is about $6.8\,\mathrm{nH}$ per square, large enough to be accessible through a calibrated microwave resonator.

Exercise 4: fermionic antisymmetry

For the uniform even-frequency setting above, suppose a pair is a spin singlet, valley triplet, and even under layer exchange. What momentum parity is required?

Solution

The spin-singlet factor is antisymmetric. Valley triplet and layer-even factors are symmetric. Their internal product is therefore antisymmetric. The momentum factor must be even so that exchanging the two complete electron labels leaves the pair amplitude antisymmetric. If the valley state were instead antisymmetric while the other labels stayed the same, odd momentum parity would be required.

Exercise 5: interpreting a resistance drop

A device shows a 90% resistance drop, a nonlinear current–voltage curve, and suppression by perpendicular field. No zero-resistance floor, stiffness, diamagnetism, or Josephson response is reported. State the strongest justified claim and two priority measurements.

Solution

The data show a field-sensitive transport anomaly compatible with superconductivity. Nonlinear current–voltage response alone does not prove collective phase coherence, so they do not yet establish a homogeneous phase-coherent bulk superconductor because percolation, fluctuation conductivity, heating, or contact effects remain.

Priority measurements include a lower-noise four-terminal zero-resistance test with current and thermal controls, plus a phase-rigidity probe such as microwave kinetic inductance, BKT-consistent scaling over a controlled range, diamagnetism, or Josephson interference. Spatial mapping would additionally test percolation.

Exercise 6: adjacency and causation

Across several devices, $T_c$ tracks a van Hove singularity while the correlated-insulator gap varies independently and vanishes in some superconducting samples. Which simple hypothesis is weakened, and what remains possible?

Solution

The hypothesis that the measured correlated insulator is a necessary parent of superconductivity is weakened: superconductivity survives without it, and their strengths do not covary. Pairing driven specifically by fluctuations of that order is also less economical unless those fluctuations persist without the static gap.

It remains possible that both phases share an underlying interaction scale, that a van Hove-enhanced phonon or electronic channel drives pairing, that a different fluctuating order is involved, or that band geometry controls the common trend. Discriminating these options requires mode-, symmetry-, and parameter-resolved tests.

References

• H. S. Arora et al., “Superconductivity in Metallic Twisted Bilayer Graphene Stabilized by WSe$_2$,” Nature 583, 379–384 (2020), doi:10.1038/s41586-020-2473-8.
• A. Banerjee et al., “Superfluid Stiffness of Twisted Trilayer Graphene Superconductors,” Nature 638, 93–98 (2025), doi:10.1038/s41586-024-08444-3.
• J. Bardeen, L. N. Cooper, and J. R. Schrieffer, “Theory of Superconductivity,” Physical Review 108, 1175–1204 (1957), doi:10.1103/PhysRev.108.1175.
• Y. Cao et al., “Unconventional Superconductivity in Magic-Angle Graphene Superlattices,” Nature 556, 43–50 (2018), doi:10.1038/nature26160.
• F. K. de Vries et al., “Gate-Defined Josephson Junctions in Magic-Angle Twisted Bilayer Graphene,” Nature Nanotechnology 16, 760–763 (2021), doi:10.1038/s41565-021-00896-2.
• J. C. Hoke et al., “Link between Thermodynamic Correlation Signatures and Superconductivity in Twisted Trilayer Graphene,” Nature Nanotechnology 21, 772–778 (2026), doi:10.1038/s41565-026-02155-8.
• H. Isobe, N. F. Q. Yuan, and L. Fu, “Unconventional Superconductivity and Density Waves in Twisted Bilayer Graphene,” Physical Review X 8, 041041 (2018), doi:10.1103/PhysRevX.8.041041.
• D. R. Nelson and J. M. Kosterlitz, “Universal Jump in the Superfluid Density of Two-Dimensional Superfluids,” Physical Review Letters 39, 1201–1205 (1977), doi:10.1103/PhysRevLett.39.1201.
• J. Pearl, “Current Distribution in Superconducting Films Carrying Quantized Fluxoids,” Applied Physics Letters 5, 65–66 (1964), doi:10.1063/1.1754056.
• S. Peotta and P. Törmä, “Superfluidity in Topologically Nontrivial Flat Bands,” Nature Communications 6, 8944 (2015), doi:10.1038/ncomms9944.
• E. Portolés et al., “A Tunable Monolithic SQUID in Twisted Bilayer Graphene,” Nature Nanotechnology 17, 1159–1164 (2022), doi:10.1038/s41565-022-01222-0.
• Y. Saito et al., “Independent Superconductors and Correlated Insulators in Twisted Bilayer Graphene,” Nature Physics 16, 926–930 (2020), doi:10.1038/s41567-020-0928-3.
• M. Tanaka et al., “Superfluid Stiffness of Magic-Angle Twisted Bilayer Graphene,” Nature 638, 99–105 (2025), doi:10.1038/s41586-024-08494-7.
• F. Wu, A. H. MacDonald, and I. Martin, “Theory of Phonon-Mediated Superconductivity in Twisted Bilayer Graphene,” Physical Review Letters 121, 257001 (2018), doi:10.1103/PhysRevLett.121.257001.
• Y. Xia et al., “Bandwidth-Tuned Mott Transition and Superconductivity in Moiré WSe$_2$,” Nature 650, 585–591 (2026), doi:10.1038/s41586-025-10049-3.