Twisted graphene multilayers exhibit strongly correlated insulating states and superconductivity due to the presence of ultraflat bands near the charge neutral point. In this paper, the response of ultraflat bands to lattice relaxation and a magnetic field in twisted trilayer graphene (tTLG) with different stacking arrangements is investigated by using a full tight-binding model. We show that lattice relaxations are indispensable for understanding the electronic properties of tTLG, in particular, of tTLG in the presence of mirror symmetry. Lattice relaxations renormalize the quasiparticle spectrum near the Fermi energy and change the localization of higher energy flat bands. Furthermore, different from the twisted bilayer graphene, the Hofstadter butterfly spectrum can be realized at laboratory accessible strengths of magnetic field. Our work verifies tTLG as a more tunable platform than the twisted bilayer graphene in strongly correlated phenomena.
We construct a continuum model of twisted trilayer graphene using {\it ab initio} density-functional-theory calculations, and apply it to address twisted trilayer electronic structure. Our model accounts for moiré variation in site energies, hopping between outside layers and within layers. We focus on the role of a mirror symmetry present in ABA graphene trilayers with a middle layer twist. The mirror symmetry is lost intentionally when a displacement field is applied between layers, and unintentionally when the top layer is shifted laterally relative to the bottom layer. We use two band stru…
Layer stacking and crystal lattice symmetry play important roles in the band structure and the Landau levels of multilayer graphene. ABA-stacked trilayer graphene possesses mirror-symmetry-protected monolayer-like and bilayer-like band structures. Broken mirror symmetry by a perpendicular electric field therefore induces hybridization between these bands and various quantum Hall phases emerge. We experimentally explore the evolution of Landau levels in ABA-stacked trilayer graphene under electric field. We observe a variety of valley and orbital dependent Landau level evolutions. These evoluti…
Twisted trilayer graphene is a particularly promising moiré superlattice system, due to its tunability, strong superconductivity, and complex electronic symmetry breaking. Motivated by these properties, we study lattice relaxation and the long-wavelength phonon modes of this system. We show that mirror-symmetric trilayer graphene hosts, aside from the conventional acoustic phonon modes, two classes of shear modes, which are even and odd under mirror reflection. The mirror-even modes are found to be gapless and equivalent to the "phason" modes of twisted bilayer graphene, with appropriately res…
We study the presence of higher-order van Hove singularities in mirror-symmetric twisted trilayer graphene. This geometry has recently emerged experimentally as a fascinating playground for studying correlated and exotic superconducting phases. We find that the trilayer hosts a zero-energy higher-order van Hove singularity with an exponent -1/3. The singularity is protected by the threefold rotation symmetry and a combined mirror-particle-hole symmetry and it can be tuned with only the twist angle and a perpendicular electric field. It arises from the combined merging of van Hove singularities…
Moiré superlattices have recently emerged as a novel platform where correlated physics and superconductivity can be studied with unprecedented tunability. Although correlated effects have been observed in several other moiré systems, magic-angle twisted bilayer graphene (MATBG) remains the only one where robust superconductivity has been reproducibly measured. Here we realize a new moiré superconductor, mirror symmetric magic-angle twisted trilayer graphene (MATTG) with dramatically richer tunability in electronic structure and superconducting properties. Hall effect and quantum oscillations m…
We calculate the electronic band structure of ABA-stacked trilayer graphene in the presence of external gates, using a self-consistent Hartree approximation to take account of screening. In the absence of a gate potential, there are separate pairs of linear and parabolic bands at low energy. A gate field perpendicular to the layers breaks mirror reflection symmetry with respect to the central layer and hybridizes the linear and parabolic low-energy bands, leaving a chiral Hamiltonian essentially different from that of monolayer or bilayer graphene. Using the self-consistent Born approximation,…
The effects of topology and electron-electron interactions on the phase diagram of ABC stacked trilayer graphene (TLG) at the neutrality point are investigated within a weak coupling renormalization group approach. We find that the leading instability of TLG with only a forward scattering density-density interaction is towards a mirror-breaking gapless state. Addition of a small, but finite back scattering favors gapped phases, allowing us to make connections to the existing experiments on TLG. We identify a fundamental symmetry difference between TLG and bilayer graphene (BLG) which is respon…
The magneto-optical spectra of ABC-stacked trilayer graphene are enriched by the electric field. A lot of prominent absorption peaks, which arise from the inter- Lnadau-level transitions, gradually change from the twin-peak structures into the double-peak ones with the increasing electric-field strength. This comes from the de- struction in mirror symmetry of xy-plane and the non-equivalence for two sublattices with the identical projections. Specially, a single threshold peak becomes a double- peak structure, owing to the Fermi-Dirac distribution. The absorption frequencies continuously grow …
The electronic tunneling properties of the two stable forms of trilayer graphene (TLG), rhombohedral ABC and Bernal ABA, are examined for pn and pnp junctions as realized by using a single gate (SG) or a double gate (DG). For the rhombohedral form, due to the chirality of the electrons, the Klein paradox is found at normal incidence for SG devices while at high energy interband scattering between additional propagation modes can occur. The electrons in Bernal ABA TLG can have a monolayer- or bilayer-like character when incident on a SG device. Using a DG however both propagation modes will cou…
Magic-angle twisted trilayer graphene (MATTG) recently exhibited robust superconductivity at a higher transition temperature (TC) than the bilayer version. With electric gating from both the top and bottom sides, the superconductivity was found to be closely associated to two conditions: the finite broken mirror symmetry and carrier concentrations between two to three carriers per moiré unite cell. Both conditions may be achieved by graphene-metal contacts where charge transfers and interfacial electric fields are generated to balance work function mismatch. In this study, we explore the super…
Twistronics heterostructures provide a novel route to control the electronic single particle velocity and thereby to engineer strong effective interactions. Here we show that the reverse may also hold, i.e. that these interactions strongly renormalize the band structure. We demonstrate this mechanism for mirror-symmetric magic angle twisted trilayer graphene at charge neutrality and in the vicinity of a phase transition which can be described by an Ising Gross-Neveu critical point corresponding, e.g., to the onset of valley Hall or Hall order. While the non-interacting model displays massless …
Quantum criticality and tunable Griffiths phase in superconducting twisted trilayer graphene
Rutgers the State University of New Jersey（新泽西州立罗格斯大学）；Center for Materials Theory, Rutgers University（罗格斯大学材料理论中心）；National Institute for Materials Science（国立物质材料研究所）；Flatiron Institute（平 insted 研究所）
In a recent experiment [Lin et al., arXiv:2112.07841], the superconducting phase hosted by a heterostructure of mirror-symmetric twisted trilayer graphene and WSe$_2$ was shown to exhibit significantly different critical currents in opposite directions in the absence of external magnetic fields. We here develop a microscopic theory and analyze necessary conditions for this zero-field superconducting diode effect. Taking into account the spin-orbit coupling induced in trilayer graphene via the proximity effect, we classify the pairing instabilities and normal-state orders and derive which combi…
Moiré superlattices that consist of two or more layers of two-dimensional materials stacked together with a small twist angle have emerged as a tunable platform to realize various correlated and topological phases, such as Mott insulators, unconventional uperconductivity and quantum anomalous Hall effect. Recently, the magic-angle twisted trilayer graphene (MATTG) has shown both robust superconductivity similar to magic-angle twisted bilayer graphene (MATBG) and other unique properties, including the Pauli-limit violating and re-entrant superconductivity. These rich properties are deeply roote…
Mirror-symmetric magic-angle twisted trilayer graphene (MATTG) hosts flat electronic bands close to zero energy, and has been recently shown to exhibit abundant correlated quantum phases with flexible electrical tunability. However studying these phases proved challenging as these are obscured by intertwined Dirac bands. In this work, we demonstrate a novel spectroscopy technique, that allows to quantify the energy gaps and Chern numbers of the correlated states in MATTG by driving band crossings between Dirac cone Landau levels and the energy gaps in the flat bands. We uncover hard correlated…
We derive the Hamiltonian for trilayer moiré systems with the Coulomb interaction projected onto the bands near the charge neutrality point. Motivated by the latest experimental results, we focus on the twisted symmetric trilayer graphene (TSTG) with a mirror-symmetry with respect to the middle layer. We provide a full symmetry analysis of the non-interacting Hamiltonian with a perpendicular displacement field coupling the band structure made otherwise of the twisted bilayer graphene (TBG) and the high velocity Dirac fermions, and we identify a hidden non-local symmetry of the problem. In the …
We study the energetic stability and the local electronic structure of the general twisted trilayer graphene (TTG) with the top and bottom layers rotated with respect to the middle layer respectively by $θ$ and $θ'$. Approximate supercells of the moiré-of-moiré superlattices with $θ$ and $θ^{\prime}$ within $1^{\circ}\sim 2^{\circ}$ are established to describe the structural and electronic properties of relaxed TTG with the periodic boundary condition. Full relaxation demonstrates that the commensurate TTG with $θ=θ^{\prime}$ has the local minimum total energy ($E_{tol}$) at a fixed $θ$, while…
Magic-angle twisted trilayer graphene (MATTG) has emerged as a novel moiré material that exhibits both strong electronic correlations and unconventional superconductivity. However, spectroscopic studies of its electronic properties are lacking, and the nature of superconductivity and the corresponding order parameter in this system remain elusive. Here we perform high-resolution scanning tunneling microscopy and spectroscopy of MATTG and reveal extensive regions of atomic reconstruction that favor mirror-symmetric stacking. In these regions, we observe a cascade of symmetry-breaking electronic…
We study theoretically a moiré superlattice geometry consisting of mirror-symmetric twisted trilayer graphene surrounded by identical transition metal dichalcogenide layers. We show that this setup allows to switch on/off and control the spin-orbit splitting of the Fermi surfaces via application of a perpendicular displacement field $D_0$, and explore two manifestations of this control: first, we compute the evolution of superconducting pairing with $D_0$; this features a complex admixture of singlet and triplet pairing and, depending on the pairing state in the parent trilayer system, phase t…
Due to the observed superconductivity, the alternating twisted trilayer graphene (ATTLG) has drawn great research interest very recently, in which three monolayer graphene (MLG) are stacked in alternating twist way. If one or several of the MLG in ATTLG are replaced by a multilayer graphene, we get a double twisted multilayer graphene (DTMLG). In this work, we theoretically illustrate that, if the DTMLG has a mirror symmetry along z direction like the ATTLG, there exists a mirror symmetry decomposition (MSD), by which the DTMLG can be exactly decoupled into two subsystems with opposite parity.…
Lattice dynamics play a crucial role in the physics of Moiré systems. In twisted bilayer graphene (TBG), it was shown that, in addition to the graphene phonons, there is another set of gapless excitations termed Moiré Phonons [Phys. Rev. B, 075416, 2019] reflecting the lattice dynamics at the Moire superlattice level. These modes were later suggested to be phasons due to the incommensurate stacking of the two graphene layers [Phys. Rev. B, 155426, 2019]. In this work, we elucidate the equivalence of these two seemingly distinct perspectives by identifying an underlying symmetry, which we dub m…
Twisted graphene systems have draw significant attention due to the discoveries of various correlated and topological phases. In particular, recently the alternating twisted trilayer graphene is discovered to exhibit unconventional superconductivity, which motivates us to study the electronic structures and possible interesting correlation effects of this class of alternating twisted graphene systems. In this work we consider generic alternating twisted multilayer graphene (ATMG) systems with $M$-$L$-$N$ stacking configurations, in which the $M$ ($L$) graphene layers and the $L$ ($N$) layers a…
The celebrated phenomenon of quantum Hall effect has recently been generalized from transport of conserved charges to that of other approximately conserved state variables, including spin and valley, which are characterized by spin- or valley-polarized boundary states with different chiralities. Here, we report a new class of quantum Hall effect in ABA-stacked graphene trilayers (TLG), the quantum parity Hall (QPH) effect, in which boundary channels are distinguished by even or odd parity under the systems mirror reflection symmetry. At the charge neutrality point and a small perpendicular mag…
