Twisted Trilayer Graphene

• Twisted trilayer graphene is a van der Waals heterostructure of three graphene layers with independent twist angles that produce intertwined moiré patterns and complex domain structures.
• Its electronic structure features nearly flat bands, van Hove singularities, and nontrivial topological states that give rise to correlated phases such as superconductivity and orbital magnetism.
• External tuning via electric fields and hydrostatic pressure allows precise control over stacking order and quantum phase transitions, making TTG a unique platform for twistronics.

Twisted trilayer graphene (TTG) is a van der Waals heterostructure composed of three graphene monolayers stacked with two independent twist angles. Compared to its bilayer counterparts, TTG exhibits a much richer array of symmetry, atomic reconstruction, and electronic correlation phenomena. The independent tuning of both twist angles produces multiple intertwined moiré superlattices and yields multi-scale structure in both real and reciprocal space. These features enable a range of correlated electronic phases—including unconventional superconductivity, orbital magnetism, topological flat bands, and spatially modulated charge order—making TTG a prototype for the paper of strong correlations and topological order in moiré materials.

1. Lattice Stacking, Moiré Geometry, and Atomic Relaxation

In TTG, each graphene layer can be independently rotated, giving twist angles θ₁₂ and θ₂₃ between adjacent pairs (layers 1–2 and 2–3). This leads to two primary moiré lattice periods, λ{12} and λ{23}, related to each angle by λ{ij} = a / [2 sin(|θ{ij}| / 2)], where a = 0.246 nm is the graphene lattice constant. When the twist angles are similar and mirror-symmetric (θ₁₂ ≈ θ₂₃), the system forms locally periodic moiré patterns with domain structures that reconstruct at small angles due to lattice relaxation (Lin et al., 2022, Park et al., 24 Feb 2024).

Atomic relaxation plays a central role, concentrating strain into networks of domain walls and producing nontrivial tessellations—triangular, kagome, vortex, or complex hexagonal networks—depending on the specific twist angles and their proximity to commensurate configurations (Park et al., 24 Feb 2024). The energetically favored stacking often alternates between Bernal-like (ABA/ACA; lower energy) and rhombohedral (ABC/ACB; slightly higher energy) orders, with the domain pattern determined by a balance between stacking energies and the cost of forming domain walls. The resulting real-space structure, especially at small twist angles, involves a large-scale "moiré-of-moiré" (MoM) superlattice that sets the localization and symmetry of electronic states.

2. Electronic Structure: Flat Bands, van Hove Singularities, and Topology

The low-energy band structure of TTG is controlled by the two moiré patterns and the degree of lattice relaxation. At generic twist angles, the band structure features van Hove singularities (VHS) near the Fermi level. In mirror-symmetric TTG with an AB-stacked bilayer (Bernal) plus a twisted monolayer, the spectrum combines two parabolic Bernal-like bands and a twisted-like Dirac cone, with the latter exhibiting strong Fermi velocity renormalization and an energy shift that grows as the twist angle is reduced (Morell et al., 2013). Gaps open in the parabolic bands, and VHS peaks or nearly flat bands emerge as the twist angles approach certain "magic" values (Ma et al., 2019, Zhu et al., 2020).

In more general configurations, the two independent twists create a "supermoiré" modulation, with the resulting interference and relaxation yielding quasiperiodic or polycrystalline electronic structures (Hao et al., 17 Jan 2024, Xia et al., 3 Sep 2025). The flat bands observed in these systems are highly sensitive to the local stacking symmetry, and theoretical models confirm that an infinite set of twist angle pairs—not just a single magic angle—can yield nearly flat bands with enhanced density of states and strong electronic correlations. The flat band formation is a haLLMark of suppressed kinetic energy, and in TTG can occur for angle pairs (θ{TM}, θ{MB}) corresponding to various commensurate/incommensurate configurations.

Topologically, TTG is predicted to host flat bands carrying nonzero valley Chern numbers, with the exact value depending on θ and an applied displacement field E_perp (Ma et al., 2019, Imran et al., 2022). These topologically nontrivial flat bands enable the emergence of exotic phases such as quantum parity Hall states, where mirror symmetry protects sets of counterpropagating edge modes with dissipationless combined Hall and longitudinal resistance quantizations.

3. Correlated Electronic Phases: Superconductivity, Magnetism, and Insulating Order

The suppression of the Fermi velocity and the presence of VHSs or flat bands in TTG substantially enhance the importance of electron-electron interactions. Several classes of correlated phases have been identified:

• Superconductivity: Robust superconductivity is observed near magic twist angles (e.g., ~1.12°, 1.35°, 1.57°), with critical temperatures up to 2.9 K reported (Turkel et al., 2021, Lake et al., 2021, Hoke et al., 9 Sep 2025). Experimental and theoretical analyses show that the superconducting order parameter is not purely spin-singlet but contains an admixture of singlet and triplet components, allowing the superconductor to exceed the Pauli limit by large factors and to survive strong in-plane magnetic fields (Lake et al., 2021). In some cases, quantum Lifshitz transitions occur at high fields, leading to reentrant superconductivity with finite-momentum pairing.
• Correlated Insulators: Measurements of local thermodynamic compressibility using scanning SET reveal gapped correlated insulators at integer fillings (e.g., ν = ±2), with the gap (Δ) exhibiting strong electron-hole asymmetry and depending sensitively on local twist angle (Hoke et al., 9 Sep 2025). The sawtooth modulation of dμ/dn in compressibility, a thermodynamic signature of strong correlations, is found to closely track the optimal conditions for superconductivity, highlighting their linked origin in the flat bands.
• Orbital Magnetism and Valley Order: At intermediate twist angles and near charge neutrality, TTG shows evidence for orbital magnetism driven by spontaneous valley polarization, distinct from the anomalous Hall effect in bilayer systems (Bhardwaj et al., 15 Dec 2024). This orbital magnetism is tunable by displacement field and carrier density and competes with, or is suppressed by, superconductivity at larger fillings.
• Quantum Griffiths Phase and Superconductor-Insulator Transition: Disorder and quantum phase fluctuations lead to superconductor-insulator transitions (SIT) tuned by magnetic field; in TTG, these exhibit quantum Griffiths behavior—rare superconducting regions persist within a globally insulating phase, yielding ultraslow relaxation dynamics and infinite-randomness critical scaling (Mahapatra et al., 14 Jul 2025). Control via magnetic field orientation sharpens or broadens the critical regime, highlighting the interplay of dimensionality, disorder, and strong correlations.

4. Moiré Interference, Quasicrystals, and Domain Engineering

The coexistence of two independent moiré superlattices in TTG allows for the engineering of a wide variety of electronic structures:

• Commensurate/Incommensurate Supermoiré Patterns: Interference between the moiré patterns leads to real-space domains whose periodicity can range from tens to hundreds of nanometers (Lin et al., 2022, Meng et al., 2022, Ren et al., 2023). These patterns underpin the emergence of supermoiré lattices, quasicrystals, and polycrystalline domain structures. In quasicrystalline TTG, local stacking varies smoothly, resulting in spatially modulated superconductivity and distinct features in the local density of states (Hao et al., 17 Jan 2024, Xia et al., 3 Sep 2025). In polycrystalline configurations, sharp domain boundaries separate regions of locally commensurate stacking, and broken xy-inversion symmetry in these domains leads to the observation of anomalous Hall effects.
• Emergent Superstructures: Double-moiré interference can create novel real-space superstructures such as Kagome lattices or hexagonal domain networks, supporting electronic states with unusual topology and localization (Meng et al., 2022, Park et al., 24 Feb 2024). Brown–Zak oscillations and inter-moiré Hofstadter butterflies have been experimentally observed at intermediary quasicrystal length scales, matching the reconstructed MoM unit cells (Ren et al., 2023).
• Atomic Relaxation and Domain Tessellation: Transmission electron microscopy and atomistic simulations reveal a rich structural phase diagram where TTG tessellates into triangular, kagome, hexagram, or vortex domain patterns, each bounded by a network of domain walls. The area fractions of Bernal and rhombohedral stacking orders jump discontinuously at specific twist angles, marking phase transitions accompanied by symmetry breaking or nematic electronic order (Park et al., 24 Feb 2024).

5. Manipulation and Control: Electric Fields, Pressure, and Device Engineering

• Electric Displacement Field: Out-of-plane electric fields break mirror symmetry, enabling hybridization of otherwise orthogonal parity bands and inducing significant changes to band topology and correlated states. The application of displacement fields yields isolated, weakly dispersive flat bands at charge neutrality—localized primarily on the middle layer—and strongly affects quantum Hall topology and phase transitions (Imran et al., 2022, Davydov et al., 13 Apr 2025).
• Hydrostatic Pressure: Vertical pressure compresses the interlayer distance, increasing the interlayer coupling. This allows the magic angle for flat-band formation to be tuned upwards, making it possible to realize flat bands and correlated states even in devices where twist angles exceed the "ambient" magic value (Wu et al., 2021). Pressure also modifies plasmon modes, with high-energy undamped and low-energy flat-band plasmons tunably present in the system.
• Electrostatic Tuning and Josephson Junctions: Local gates permit electrostatic control over the quantum phase transitions in device geometries, as in gate-defined Josephson junctions used to investigate competing superconducting and magnetic orders (Bhardwaj et al., 15 Dec 2024). Fine adjustment of the weak link enables switching between superconducting and orbital magnetic phases, with signatures like asymmetric Fraunhofer interference revealing valley-polarized orbital ferromagnetism.

6. Topological Order, Quantum Hall States, and Layer Polarization

TTG exhibits unconventional layer-polarized quantum Hall states governed by the intricate interplay of interlayer coupling, screening, and charge redistribution. Each layer can host Landau levels with independent Chern numbers, and transitions (intra- and interlayer) between ordered phases are controlled by carrier density and displacement field (Davydov et al., 13 Apr 2025). Sharp features in the stability diagram map to changes in layer filling configurations, leading to complex transport phenomena, including possible excitonic condensates at domain boundaries. These states are sensitive to both the position within the stability diagram and to external fields, highlighting the potential for device engineering of arbitrary correlated and topological phases in moiré superlattices.

7. Outlook and Future Directions

Twisted trilayer graphene, with its tunable multi-moiré structure, domain engineering, and strong correlation physics, stands as a paradigmatic system for the realization of novel quantum phases. The observed magic continuum in twist-angle space (Xia et al., 3 Sep 2025) shows that correlated flat-band phenomena and associated superconductivity, magnetism, or topological order are robust across entire manifolds—not just isolated "magic angles." The, now, demonstrated ability to probe and control the interplay of stacking order, twist-angle disorder, atomic relaxation, electric field, and pressure paves the way for future advances in twistronic device design, reconfigurable moiré qubits, exotic phase transitions, and the exploration of quantum criticality in complex correlated materials.