In small-angle twisted bilayer graphene (TBLG), transport measurements on a twist angle θ≈1.8∘\theta\approx1.8^\circ device (Cao et al., Phys. Rev. Lett. 117, 116804 (2016)) revealed insulating states at the moiré superlattice band edges with thermal activation gaps of roughly 50–60 meV, several times larger than the gaps predicted by single-particle band-structure calculations, together with Shubnikov–de Haas oscillations showing two valley-polarized Fermi contours whose crossing points are effectively unhybridized. The open problem is to give a quantitative, mechanism-resolved microscopic account of these observations: to determine which combination of underestimated interlayer hopping and interaction strength, lattice relaxation and heterostrain, and many-body (e.g. excitonic) effects produces the measured gaps (including the hole-side insulating state that early calculations missed), to establish the microscopic origin of the non-activated transport below ∼\sim50 K, to determine the magnitude and momentum dependence of the interlayer hopping matrix elements in realistic relaxed structures, and to establish whether residual hybridization at the valley-protected crossings is finite and whether magnetic breakdown affects the observed quantum-oscillation orbits. Lattice-relaxation theory and the post-2018 magic-angle correlated-insulator literature address parts of this, but no quantitative reconciliation exists for non-magic small twist angles.

Background

Twisted bilayer graphene (TBLG, also written TwBLG) consists of two graphene layers rotated by a relative twist angle θ\theta; the mismatch produces a moiré pattern with period λSL\lambda_{\mathrm{SL}} and a mini Brillouin zone (MBZ, the folded Brillouin zone of the superlattice) that folds the bands of both layers. For θ>3∘\theta>3^{\circ} the layers are effectively decoupled, while for small θ\theta interlayer tunneling acts as a periodic Bragg potential that reconstructs the bands; the standard continuum description is due to Bistritzer and MacDonald ('Moiré butterflies in twisted bilayer graphene', Phys. Rev. B 84, 035440 (2011)). At full filling of the lowest miniband with its fourfold spin–valley degeneracy (four carriers per moiré unit cell), band-structure calculations predict energy gaps at the Γs\Gamma_s point of the MBZ, of order a few to roughly ten meV in the relevant angle range.

Cao, Luo, Fatemi, Fang, Sanchez-Yamagishi, Watanabe, Taniguchi, Kaxiras and Jarillo-Herrero (Phys. Rev. Lett. 117, 116804 (2016)) measured dual-gated, hexagonal-boron-nitride (hBN)-encapsulated TBLG devices with θ≈1.8∘\theta\approx1.8^{\circ}. They observed insulating states at total densities n≈±7.5×1012 cm−2n\approx\pm7.5\times10^{12}\,\mathrm{cm}^{-2} on both the electron and the hole side, with slopes of fits to the activated (Arrhenius) temperature dependence between 100 K and 300 K giving activation gaps of about 50 meV (electron side) and 60 meV (hole side); the conductivity saturates below ∼\sim50 K, which the authors tentatively attributed to variable-range hopping (transport by thermally activated tunneling between disorder-localized states, whose rate grows as a stretched exponential of temperature). Two discrepancies with the ab initio tight-binding calculations used in that work were flagged: the calculations predicted no global transport gap on the hole side, and the calculated electron-side gap was far smaller than measured. The authors listed candidate mechanisms — an underestimated interlayer interaction strength in the calculations (deemed unlikely to explain most of the difference), lattice strain effects analogous to those proposed for graphene on hexagonal boron nitride, and an excitonic instability (spontaneous condensation of electron–hole pairs into an insulating state, as discussed for Bernal-stacked (conventionally AB-stacked) bilayer graphene).

The same measurements analyzed Shubnikov–de Haas (SdH) oscillations, quantum oscillations of the longitudinal resistivity whose frequency encodes extremal Fermi-surface areas. Near the insulating states a single oscillation frequency with fourfold degeneracy was found, matching only the triangular orbits of the calculated Star-of-David-shaped Fermi contours. The two relevant Fermi contours are valley polarized (derived from the KK points of one layer and the K′K' points of the other), and although they intersect inside the MBZ they remain effectively unhybridized: the interlayer hopping amplitude decays exponentially with the momentum mismatch of the originating monolayer states, on a characteristic scale given by the moiré wave vector kSLk_{\mathrm{SL}}, and hybridizing the crossing states requires an inter-valley momentum transfer much larger than kSLk_{\mathrm{SL}}. The authors noted (in a footnote) the alternative possibility that any residual hybridization at these crossings is so small that magnetic breakdown — quantum tunneling between semiclassical kk-space orbits at near-degeneracy points, as treated by Lu and Fertig for TBLG (Phys. Rev. B 89, 085408 (2014)) — has already occurred in the fields used. Determining whether the crossings carry a finite (sub-resolution) hybridization gap, and whether magnetic breakdown affects the observed orbits, requires quantum-oscillation or spectroscopic probes beyond the original experiment.

Later work has clarified pieces of this picture. Lattice relaxation, i.e. the spontaneous reconstruction of the twisted stack into alternating AB/BA-stacking domains with shrunken AA regions, was shown by Nam and Koshino to open gaps of up to about 20 meV at the superlattice subband edges on both electron and hole sides — qualitatively explaining why a hole-side insulating state exists at all, but leaving the measured 50–60 meV gaps only partially accounted for. Quantitative strain mapping has established that heterostrain (a uniform strain of one layer relative to the other) and twist-angle disorder are substantial in real devices, and the magic-angle correlated-insulator literature has demonstrated that interactions can open large gaps in TBLG, but at integer fillings of the flat band near 1.1∘1.1^{\circ}, a different regime from the single-particle band-edge gaps of a ∼2∘\sim2^{\circ} device. What is still missing is a microscopic model that quantitatively reproduces the measured gap sizes, bandwidths, and quantum-oscillation geometry of small-angle TBLG with an identified decomposition of mechanisms.

Problem Statement

Give a quantitative, mechanism-resolved microscopic account of the superlattice band-edge insulating states and the Fermi-surface topology of small-angle twisted bilayer graphene, as observed in the original θ≈1.8∘\theta\approx1.8^{\circ} transport measurements (activation gaps of roughly 50 meV on the electron side and 60 meV on the hole side, several times larger than single-particle predictions, with conductivity saturating below ∼\sim50 K, and a single SdH frequency consistent with triangular valley-polarized orbits on effectively unhybridized, intersecting Fermi contours). Specifically: (a) determine which combination of underestimated interlayer hopping and interaction strength, lattice relaxation, heterostrain, and many-body (e.g. excitonic) effects produces the observed gap magnitudes, and account for the presence of the hole-side insulating state; (b) establish the microscopic origin of the departure from activated behavior below ∼\sim50 K, i.e. whether it is disorder-driven variable-range hopping or reflects intrinsic localization, and what role disorder plays generally; (c) determine the magnitude and momentum dependence of the interlayer hopping matrix elements in realistic, relaxed and possibly strained TBLG structures, including the exponential suppression scale relative to the moiré wave vector, so that calculated bandwidths, Fermi velocities, and the protection of the crossing points match experiment; and (d) establish whether the hybridization at the valley-protected Fermi-contour crossings is exactly zero or merely below experimental resolution, and whether magnetic breakdown affects the observed SdH orbits in accessible field and density regimes. These four facets are parts of a single objective: a validated microscopic model of the low-energy band structure of small-angle TBLG that quantitatively reproduces both the transport gaps and the quantum-oscillation data (the 50/60 meV activation gaps on both sides, the bandwidths and Fermi velocities, the protection of the valley-polarized crossings, and the SdH orbit geometry) to within the uncertainty of the cited measurements.

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