The discovery of magic-angle twisted trilayer graphene (tTLG) adds a new twist to the family of graphene moiré. The additional graphene layer unlocks a series of intriguing properties in the superconducting phase, such as the violation of Pauli limit and re-entrant superconductivity at large in-plane magnetic field. In this work, we integrate magic-angle tTLG into a double-layer structure to study the superconducting phase. Utilizing proximity screening from the adjacent metallic layer, we examine the stability of the superconducting phase and demonstrate that Coulomb repulsion competes against the mechanism underlying Cooper pairing. Furthermore, we use a combination of transport and thermodynamic measurements to probe the isospin order, which points towards a spin-polarized and valley-unpolarized isospin configuration at half moiré filling, and for the nearby fermi surface. Our findings provide important constraints for theoretical models aiming to understand the nature of superconductivity. A possible scenario is that electron-phonon coupling stabilizes a superconducting phase with a spin-triplet, valley singlet order parameter.
It has recently been shown that superconductivity in magic-angle twisted trilayer graphene survives to in-plane magnetic fields that are well in excess of the Pauli limit, and much stronger than the in-plane critical magnetic fields of magic-angle twisted bilayer graphene. The difference is surprising because twisted bilayers and trilayers both support the magic-angle flat bands thought to be the fountainhead of twisted graphene superconductivity. We show here that the difference in critical magnetic fields can be traced to a $\mathcal{C}_2 \mathcal{M}_{h}$ symmetry in trilayers that survives …
We consider twisted bilayer and trilayer graphene in the presence of Rashba spin-orbit coupling and explore the physics of Moiré spintronics. The electronic charge density has a sharp step right at the magic angles $θ_m$. As a result, local spin observables (polarization and equilibrium spin currents) have sharp peaks (of width about a small fraction of 1$^\circ$) as a function of the twist angle $θ$, and abrupt sign reversals at $θ_m$. Thereby, the magic angle can be determined in an unprecedented accuracy. In the first chiral limit, the spin currents vanish, but the peculiar pattern of the p…
We consider a configuration of three stacked graphene monolayers with equal consecutive twist angles $θ$. Remarkably, in the chiral limit when interlayer coupling terms between $\textrm{AA}$ sites of the moiré pattern are neglected we find four perfectly flat bands (for each valley) at a sequence of magic angles which are exactly equal to the twisted bilayer graphene (TBG) magic angles divided by $\sqrt{2}$. Therefore, the first magic angle for equal-twist trilayer graphene (eTTG) in the chiral limit is $θ_{*} \approx 1.05^{\circ}/\sqrt{2} \approx 0.74^{\circ}$. We prove this relation analytic…
We calculate the static polarization of AAA-stacked trilayer graphene (TLG) and study its screening properties within the random phase approximation (RPA) in all undoped, doped and biased regimes. We find that the static polarization of undoped AAA-stacked TLG is a combination of the doped and undoped single layer graphene static polarization. This leads to an enhancement of the dielectric background constant along a Thomas-Fermi screening with the Thomas-Fermi wave vector which is independent of carrier concentrations and a 1/r^3 power law decay for the long-distance behavior of the screened …
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…
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…
Starting with twisted bilayer graphene, graphene-based moiré materials have recently been established as a new platform for studying strong electron correlations. In this paper, we study twisted graphene monolayers on trilayer graphene and demonstrate that this system can host flat bands when the twist angle is close to the magic-angle of 1.16$^\circ$. When monolayer graphene is twisted on ABA trilayer graphene, the flat bands are not isolated, but are intersected by a Dirac cone with a large Fermi velocity. In contrast, graphene twisted on ABC trilayer graphene (denoted AtABC) exhibits a gap …
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 …
Due to its potential connection with nematicity, electronic anisotropy has been the subject of intense research effort on a wide variety of material platforms. The emergence of spatial anisotropy not only offers a characterization of material properties of metallic phases, which cannot be accessed via conventional transport techniques, but it also provides a unique window into the interplay between Coulomb interaction and broken symmetry underlying the electronic order. In this work, we utilize a new scheme of angle-resolved transport measurement (ARTM) to characterize electron anisotropy in m…
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…
In this article we study the ferromagnetic behavior of ABC-stacked trilayer graphene. This is done using a nearest-neighbor tight-binding model, in the presence of long-range Coulomb interactions. For a given electron-electron interaction g and doping level n, we determine whether the total energy is minimized for a paramagnetic or ferromagnetic configuration of our variational parameters. The g versus n phase diagram is first calculated for the unscreened case. We then include the effects of screening using a simplified expression for the fermion bubble diagram. We show that ferromagnetism in…
We investigate the influence of Meissner screening and trapped magnetic flux on magnetization dynamics for a Ni80Fe20 film sandwiched between two thick Nb layers (100 nm) using broadband (5-20 GHz) ferromagnetic resonance (FMR) spectroscopy. Below the superconducting transition Tc of Nb, significant zero-frequency line broadening (5-6 mT) and DC resonance field shift (50 mT) to a low field are both observed if the Nb thickness is comparable to the London penetration depth of Nb films (>= 100 nm). We attribute the observed peculiar behaviors to the increased incoherent precession near the Ni80F…
Magic-angle twisted trilayer graphene (MATTG) recently emerged as a highly tunable platform for studying correlated phases of matter, such as correlated insulators and superconductivity. Superconductivity occurs in a range of doping levels that is bounded by van Hove singularities which stimulates the debate of the origin and nature of superconductivity in this material. In this work, we discuss the role of spin-fluctuations arising from atomic-scale correlations in MATTG for the superconducting state. We show that in a phase diagram as function of doping ($ν$) and temperature, nematic superco…
Heavy fermion phase diagram in magic-angle twisted trilayer graphene
Westlake University（西湖大学）；Westlake Institute for Advanced Study（西湖高等研究院）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；IMDEA Nanociencia（IMDEA纳米科学中心）；National Institute for Materials Science（国立材料研究所）
Recent experimental and theoretical investigations demonstrate that twisted trilayer graphene (tTLG) is a highly tunable platform to study the correlated insulating states, ferromagnetism, and superconducting properties. Here we explore the possibility of tuning electronic correlations of the tTLG via a vertical pressure. A full tight-binding model is used to accurately describe the pressure-dependent interlayer interactions. Our results show that pressure can push a relatively larger twist angle (for instance, $1.89^{\circ}$) tTLG to reach the flat-band regime. Next, we obtain the relationshi…
Moiré quantum matter has emerged as a novel materials platform where correlated and topological phases can be explored with unprecedented control. Among them, magic-angle systems constructed from two or three layers of graphene have shown robust superconducting phases with unconventional characteristics. However, direct evidence for unconventional pairing remains to be experimentally demonstrated. Here, we show that magic-angle twisted trilayer graphene (MATTG) exhibits superconductivity up to in-plane magnetic fields in excess of 10 T, which represents a large ($2\sim3$ times) violation of th…
We construct a van der Waals heterostructure consisting of three graphene layers stacked with alternating twisting angles $\pmθ$. At the average twist angle $θ\sim 1.56^{\circ}$, a theoretically predicted magic angle for the formation of flat electron bands, narrow conduction and valence moiré bands appear together with a linearly dispersing Dirac band. Upon doping the half-filled moiré valence band with holes, or the half-filled moiré conduction band with electrons, displacement field tunable superconductivity emerges, reaching a maximum critical temperature of 2.1 K at optimal doping and dis…
In this article we study the static polarization in ABC-stacked multilayer graphene. Since the density of states diverges for these systems if the number of layers exceeds three, screening effects are expected to be important. In the random phase approximation, screening can be included through the polarization. We derive an analytical integral expression for the polarization in both the full-band model and an effective two-band model. Numerical evaluation of these integrals are very time consuming in the full-band model. Hence, for ABC-stacked trilayer graphene, we use the two-band model to c…
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…
Trilayer graphene with a twisted middle layer has recently emerged as a new platform exhibiting correlated phases and superconductivity near its magic angle. A detailed characterization of its electronic structure in the parameter space of twist angle $θ$, interlayer potential difference $Δ$, and top-bottom layer stacking $τ$ reveals that flat bands with large Coulomb energy vs bandwidth $U/W > 1$ are expected within a range of $\pm 0.2^{\circ}$ near $θ\simeq1.5^{\circ}$ and $θ\simeq1.2^{\circ}$ for $τ_{\rm AA}$ top-bottom layer stacking, between a wider $1^{\circ} \sim 1.7^{\circ}$ range for …
Kekulé spirals and charge transfer cascades in twisted symmetric trilayer graphene
Princeton Center for Theoretical Science, Princeton University（普林斯顿大学理论科学中心）；Department of Physics, University of Zurich（苏黎世大学物理系）；Department of Physics, Ghent University（根特大学物理系）
Magic angle twisted trilayer graphene (TTG) has recently emerged as a new platform to engineer strongly correlated flat bands. Here, we reveal the structural and electronic properties of TTG using low temperature scanning tunneling microscopy at twist angles for which superconductivity has been observed. Real trilayer samples deviate from their idealized structure due to a strong reconstruction of the moiré lattice, which locks layers into near-magic angle, mirror symmetric domains comparable in size to the superconducting coherence length. The price for this magic relaxation is the introducti…
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,…
We uncover a new type of magic-angle phenomena when an AA-stacked graphene bilayer is twisted relative to another graphene system with band touching. In the simplest case this constitutes a trilayer system formed by an AA-stacked bilayer twisted relative to a single layer of graphene. We find multiple anisotropic Dirac cones coexisting in such twisted multilayer structures at certain angles, which we call "Dirac magic." We trace the origin of Dirac magic angles to the geometric structure of the twisted AA-bilayer Dirac cones relative to the other band-touching spectrum in the moiré reciprocal …
We introduce twisted trilayer graphene (tTLG) with two independent twist angles as an ideal system for the precise tuning of the electronic interlayer coupling to maximize the effect of correlated behaviors. As established by experiment and theory in the related twisted bilayer graphene system, van Hove singularities (VHS) in the density of states can be used as a proxy of the tendency for correlated behaviors. To explore the evolution of VHS in the twist-angle phase space of tTLG, we present a general low-energy electronic structure model for any pair of twist angles. We show that the basis o…
Twisted trilayer graphene (TLG) may be the simplest realistic system so far, which has flat bands with nontrivial topology. Here, we give a detailed calculation about its band structures and the band topology, i.e. valley Chern number of the nearly flat bands, with the continuum model. With realistic parameters, the magic angle of twisted TLG is about 1.12$^{\circ}$, at which two nearly flat bands appears. Unlike the twisted bilayer graphene, a small twist angle can induce a tiny gap at all the Dirac points, which can be enlarged further by a perpendicular electric field. The valley Chern numb…
Motivated by the recent experimental realization of twisted trilayer graphene and the observed superconductivity that is associated with its flat bands at specific angles, we study trilayer graphene under the influence of different forms of light in the non-interacting limit. Specifically, we study four different types of stacking configurations with a single twisted layer. In all four cases, we study the impact of circularly polarized light and longitudinal light coming from a waveguide. We derive effective time-independent Floquet Hamiltonians and review light-induced changes to the band str…
We have investigated the electronic structure of charged bilayer and trilayer phoshporene using first-principles, density-functional-theory calculations. We find that the effective dielectric constant for an external electric field applied perpendicular to phosphorene layers increases with the charge density and is twice as large as in an undoped system if the electron density is around $5\times10^{13}$cm$^{-2}$. It is known that if few-layer phosphorene is placed under such an electric field, the electron band gap decreases and if the strength of the electric field is further increased, the b…
We study the symmetries of twisted trilayer graphene's band structure under various extrinsic perturbations, and analyze the role of long-range electron-electron interactions near the first magic angle. The electronic structure is modified by these interactions in a similar way to twisted bilayer graphene. We analyze electron pairing due to long-wavelength charge fluctuations, which are coupled among themselves via the Coulomb interaction and additionally mediated by longitudinal acoustic phonons. We find superconducting phases with either spin singlet/valley triplet or spin triplet/valley sin…
The ability to control the strength of interaction is essential for studying quantum phenomena emerging from a system of correlated fermions. For example, the isotope effect illustrates the effect of electron-phonon coupling on superconductivity, providing an important experimental support for the BCS theory. In this work, we report a new device geometry where the magic-angle twisted bilayer graphene (tBLG) is placed in close proximity to a Bernal bilayer graphene (BLG) separated by a 3 nm thick barrier. Using charge screening from the Bernal bilayer, the strength of electron-electron Coulomb …
Coulomb screening of superconductivity in magic-angle twisted bilayer graphene
School of Physics & Astronomy, University of Manchester（曼彻斯特大学物理与天文学学院）；National Graphene Institute, University of Manchester（曼彻斯特大学国家石墨烯研究所）；Department of Physics, Washington University in St. Louis（华盛顿大学圣路易斯分校物理系）；Department of Physics and Astronomy, University of Pennsylvania（宾夕法尼亚大学物理与天文学系）；Department of Physics, University of Antwerp（安特卫普大学物理系）；Henry Royce Institute for Advanced Materials, The University of Manchester（曼彻斯特大学亨利·罗伊斯先进材料研究所）；National Institute for Materials Science（国立材料研究所）；Department of Materials Science and Engineering, National University of Singapore（新加坡国立大学材料科学与工程系）；Department of Physics, National University of Singapore（新加坡国立大学物理系）
We show that the superconducivity in twisted graphene multilayers originates from a common feature, which is the strong valley symmetry breaking characteristic of these moiré systems at the magic angle. This leads to a breakdown of the rotational symmetry of the flat moiré bands down to $C_3$, and to ground states in which the time-reversal symmetry is broken for a given spin projection. However, this symmetry can be recovered upon exchange of spin-up and spin-down electrons, as we illustrate by means of a self-consistent microscopic Hartree-Fock resolution where the states for the two spin pr…
When layers of graphene are rotationally misaligned by the magic angle, the moiré superlattice features extremely flat bands. Due to the enhanced density of states, the Coulomb interaction induces a variety of instabilities. The most prominent occur at integer filling and are therefore commonly attributed to spontaneous polarization of the moiré unit cell's `flavor' degrees of freedom -- spin, valley, and the flat-band degeneracy. As the dominant member of the hierarchy, these correlated states are thought to crucially determine further instabilities at lower energy scales, such as superconduc…
Magic-angle twisted bilayer graphene (MA-tBLG) has appeared as a tunable testing ground to investigate the conspiracy of electronic interactions, band structure, and lattice degrees of freedom to yield exotic quantum many-body ground states in a two-dimensional Dirac material framework. While the impact of external parameters such as doping or magnetic field can be conveniently modified and analyzed, the all-surface nature of the quasi-2D electron gas combined with its intricate internal properties pose a challenging task to characterize the quintessential nature of the different insulating an…
The layer-based random-phase approximation is further developed to investigate electronic excitations in tri-layer ABC-stacked graphene. All the layer-dependent atomic interactions and Coulomb interactions are included in the dynamic charge screening. There exist rich and unique (momentum, frequency)-excitation phase diagrams, in which the complex single-particle excitations and five kinds of plasmon modes, are dominated by the unusual energy bands and doping carrier densities. The latter frequently experience the significant Landau damping due to the former, leading to the coexistence/destruc…
Understanding and tuning correlated states is of great interest and significance to modern condensed matter physics. The recent discovery of unconventional superconductivity and Mott-like insulating states in magic-angle twisted bilayer graphene (tBLG) presents a unique platform to study correlation phenomena, in which the Coulomb energy dominates over the quenched kinetic energy as a result of hybridized flat bands. Extending this approach to the case of twisted multilayer graphene would allow even higher control over the band structure because of the reduced symmetry of the system. Here, we …
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 calculate the dynamical polarization function and solve a self-consistent gap equation in the random phase approximation in undoped ABC-stacked n-layer graphene. We find that the gap is maximal in trilayer graphene and decreases monotonously for $n \ge 4$ because the effects of screening in the gap equation win over those connected with the flattening of electron bands as n increases.
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…
When two monolayers of graphene are stacked with a small relative twist angle, the resulting band structure exhibits a remarkably flat pair of bands at a sequence of 'magic angles' where correlation effects can induce a host of exotic phases. Here, we study a class of related models of $n$-layered graphene with alternating relative twist angle $\pm θ$ which exhibit magic angle flat bands coexisting with several Dirac dispersing bands at the Moiré K point. Remarkably, we find that the Hamiltonian for the multilayer system can be mapped exactly to a set of decoupled bilayers at {\it different} a…
The Pauli limiting field represents fundamental magnetic field at which superconducting state collapses due to the spin-paramagnetic Cooper pair-breaking effect. Cao et al [1] reported that the upper critical field (i.e., the magnetic field at which the superconducting state collapses in experiment) in magic-angle twisted trilayer graphene (MATNG, N=3) in 2-3 times exceeds the Pauli limiting field in this material. This observation was interpreted as a violation of the Pauli-limiting field in MAT3G. Similar conclusions were recently reported by the same research group in MATNG (N=4,5) superlat…
