Twist-angle disorder and the correlated phases of magic-angle twisted bilayer graphene: intrinsic tolerance versus percolation, the many-body consequences of gradient-induced internal in-plane electric fields, and interacting theory on the measured twist-angle landscape

Quantum-Hall tomography with a nanoSQUID-on-tip showed that state-of-the-art magic-angle twisted bilayer graphene (MATBG) devices still carry a local twist-angle spread of order $0.1^\circ$, twist-angle gradients, and a network of abrupt twist steps, and that these gradients generate sizable gate-tunable internal in-plane electric fields that reorganize the quantum Hall state. The same devices nonetheless show global correlated physics, including superconductivity and symmetry-broken correlated insulators. The unresolved question is how the correlated orders tolerate this disorder: whether superconductivity and the correlated insulators are intrinsically robust to substantial local twist-angle variation, or whether zero resistance arises from percolating paths of near-optimal twist angle; what quantitative tolerance thresholds each correlated phase has; how the internal in-plane electric fields couple to interactions, screening, pairing, magnetism, and $C_3$ symmetry breaking; and how the measured spatially varying twist angle $\theta(\mathbf{r})$ can be incorporated into microscopic interacting calculations. Later experiments (scanning-tunneling and SQUID-on-tip imaging, Josephson-network transport) and theory (Anderson-theorem robustness, strain-disorder gap suppression, random-impedance-network percolation modeling) have illuminated pieces of this question but none decides the central dichotomy or supplies the thresholds, so the problem remains open.

Background

Twisted bilayer graphene (TBG) consists of two graphene layers rotated by an interlayer twist angle $\theta$; the rotation produces a moiré superlattice with period roughly inversely proportional to $\theta$. At the first magic angle $\theta \approx 1.1^\circ$ the lowest moiré minibands become exceptionally flat, and upon electrostatic filling of these bands the material develops correlated insulating states at integer fillings $\nu$ (electrons per moiré unit cell relative to charge neutrality), superconductivity, and other symmetry-broken orders. These correlated phases are unusually sensitive to $\theta$ because the flat-band bandwidth and filling scale are set by the local twist angle.

Uri, Grover and co-workers (Nature 581, 47 (2020)) mapped the local twist angle $\theta(\mathbf{r})$ in devices encapsulated in hBN (hexagonal boron nitride, an insulating crystal used as an atomically flat substrate and top gate dielectric) by SQUID-on-tip (SOT) imaging — a scanning superconducting quantum interference device, here a nanoscale pickup loop of diameter $\approx 200$–$250$ nm that images local currents and magnetization — using Landau-level tomography: in the quantum Hall regime, the spatial positions where each Landau level (the discrete cyclotron orbits of a two-dimensional electron gas in magnetic field) crosses the local chemical potential trace the local band bottom, which is a direct function of $\theta$, so the tomographic Landau-level pattern reconstructs $\theta(\mathbf{r})$ with relative precision better than $0.002^\circ$ and spatial resolution of a few moiré periods. They found that even devices with high-quality global MATBG transport features display a local twist-angle span of order $0.1^\circ$, with smooth gradients (typically $\sim 0.05^\circ/\mu\mathrm{m}$) interrupted by a network of abrupt steps resembling stacking faults.

Two physical consequences were identified. First, regions of different $\theta$ have different densities of states and hence different chemical potentials at fixed carrier density; equilibration then forces a spatially varying electrostatic potential that produces internal, essentially unscreened in-plane electric fields, gate-tunable and estimated at sizable values for measured gradients, which fragment the quantum Hall state into bulk incompressible strips (narrow insulating ribbons at integer local filling that act as internal edge channels). Second, the measured Landau-level degeneracy pattern suggested $C_3$ (threefold in-plane rotation) symmetry breaking possibly triggered by the twist gradients. On the correlated side, the authors observed that correlated physics appears in devices whose local $\theta$ histograms barely overlap, while the Landau levels near $\nu/4$ and $\nu/2$ are spatially discontinuous — evidence both for a tolerance of correlated physics to $\theta$ and for the fragility of specific correlated states to twist-angle disorder. The question of whether superconductivity and the correlated insulators are intrinsically tolerant or arise from percolation along near-optimal-$\theta$ paths was posed as unresolved.

Relevant theory context: the leading candidates for the correlated insulators at even fillings are intervalley coherent states, in which electrons coherently occupy both valleys; the Kramers intervalley coherent (K-IVC) state and the incommensurate Kekulé spiral (IKS), a close cousin favored by strain and inhomogeneity, are prominent examples. Two-dimensional superconductors in general should undergo a Berezinskii–Kosterlitz–Thouless (BKT) transition, a topological unbinding of vortex–antivortex pairs, whose stiffness temperature-dependence distinguishes nodal from fully gapped superconducting order parameters. The band-structure and quantum-Hall calculations in the 2020 work assumed locally homogeneous $\theta$; no microscopic interacting treatment of the full measured $\theta(\mathbf{r})$ landscape, with its gradients, steps, and internal fields, existed at the time, and the experimental tolerance thresholds of the different correlated phases were unknown.

Problem Statement

The common objective is to determine how the measured twist-angle disorder landscape of magic-angle twisted bilayer graphene — a local spread of order $0.1^\circ$ with smooth gradients and a network of abrupt steps — controls its correlated phases. Within this objective, the following interlinked questions, posed by the tomographic imaging work of Uri et al. (Nature 581, 47 (2020)), remain open:

(1) Tolerance versus percolation. Superconductivity and correlated insulators are observed in devices whose local twist angle varies by about $0.1^\circ$. Determine whether these correlated orders are intrinsically tolerant of substantial local $\theta$ variation, or whether the observed global correlated behavior (in particular the zero-resistance superconducting state) arises from percolating paths of near-optimal $\theta$. Establish the quantitative tolerance thresholds — the local $\theta$ spread each correlated phase can survive — for superconductivity and for the intervalley-coherent correlated insulators at the even integer fillings, namely the Kramers intervalley coherent and incommensurate Kekulé spiral states.

(2) Internal in-plane electric fields. Self-consistent single-particle simulations show that $\nabla\theta$ produces sizable, gate-tunable, essentially unscreened internal in-plane electric fields that reorganize quantum Hall states into bulk incompressible strips. The open question is how these internal in-plane fields couple to interactions and how screening behaves beyond the single-particle approximation for the correlated phases, and what consequences they have for the pairing mechanism, magnetism, symmetry breaking (including $C_3$ rotational symmetry breaking), and nonlinear and finite-temperature transport.

(3) Many-body theory on the measured landscape. Construct microscopic interacting calculations that quantitatively incorporate the spatially varying $\theta(\mathbf{r})$ — gradients and step networks — together with the induced internal fields, so as to predict how the twist-angle landscape modifies the collective orders of MATBG.

As enabling sub-goals serving the same objective: extend the tomographic mapping of $\theta(\mathbf{r})$ to the device regions that show no MATBG Landau levels within the accessible gate range (so that the full device-scale twist-angle distribution, including near-zero and large-$\theta$ regimes, is characterized), and improve the spatial resolution of the current imaging so that the widths of incompressible strips and the current distributions within them, and hence the local electric fields and gap magnitudes at the smallest scales, are determined rather than bounded by the SQUID-on-tip diameter and imaging height.

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