Combining mean-field and renormalization group analyses, here we unveil the nature of recently observed superconductivity and parent metallic states in chemically doped rhombohedral trilayer graphene, subject to external electric displacement fields ($D$) [H. Zhou, \emph{et al.}, Nature (London) {\bf 598}, 434 (2021)]. We argue that close to the charge neutrality, on site Hubbard repulsion favors layer antiferromagnet, which when combined with the $D$-field induced layer polarization, produces a spin-polarized, but valley-unpolarized half-metal, conducive to the nucleation of spin-triplet $f$-wave pairing (SC2). At larger doping valence bond order emerges as a prominent candidate for isospin coherent paramagent, boosting condensation of spin-singlet Cooper pairs in the $s$-wave channel (SC1), manifesting a "selection rule" among competing orders. Responses of these paired states to displacement and in-plane magnetic fields show qualitative similarities with experimental observation. With the onset of the quantum anomalous Hall order, the valley degeneracy of half-metal gets lifted, forming a quarter-metal at lower doping [H. Zhou, \emph{et al.}, Nature (London) {\bf 598}, 429 (2021)].
We study the emergence of superconductivity in rhombohedral trilayer graphene due purely to the long-range Coulomb repulsion. This repulsive-interaction-driven phase in rhombohedral trilayer graphene is significantly different from those found in twisted bilayer and trilayer graphenes. In the latter case, the nontrivial momentum-space geometry of the Bloch wavefunctions leads to an effective attractive electron-electron interaction; this allows for less modulated order parameters and for spin-singlet pairing. In rhombohedral trilayer graphene, we instead find spin-triplet superconductivity wit…
Trilayer graphene is receiving an increasing level of attention due to its stacking--dependent magnetoelectric and optoelectric properties, and its more robust ferromagnetism relative to monolayer and bilayer variants. Additionally, rhombohedral stacked trilayer graphene presents the possibility of easily opening a gap via either an external electric field perpendicular to the layers, or via the application of external strain. In this paper, we consider an external electric field to open a bandgap in rhombohedral trilayer graphene and study the excitonic optical response of the system. This is…
We show that in a two-dimensional electron gas with an annular Fermi surface, long-range Coulomb interactions can lead to unconventional superconductivity by the Kohn-Luttinger mechanism. Superconductivity is strongly enhanced when the inner and outer Fermi surfaces are close to each other. The most prevalent state has chiral $p$-wave symmetry, but $d$-wave and extended $s$-wave pairing are also possible. We discuss these results in the context of rhombohedral trilayer graphene, where superconductivity was recently discovered in regimes where the normal state has an annular Fermi surface. Usin…
Motivated by the observation of two distinct superconducting phases in the moiréless ABC-stacked rhombohedral trilayer graphene, we investigate the electron-acoustic-phonon coupling as a possible pairing mechanism. We predict the existence of superconductivity with the highest $T_c\sim 3$K near the Van Hove singularity. Away from the Van Hove singularity, $T_c$ remains finite in a wide range of doping. In our model, the $s$-wave spin-singlet and $f$-wave spin-triplet pairings yield the same $T_c$, while other pairing states have negligible $T_c$. Our theory provides a simple explanation for th…
Motivated by recent experiments on ABC-stacked rhombohedral trilayer graphene (RTG) which observed spin-valley symmetry-breaking and superconductivity, we study instabilities of the RTG metallic state to symmetry breaking orders. We find that interactions select the inter-valley coherent order (IVC) as the preferred ordering channel over a wide range, whose theoretically determined phase boundaries agree well with experiments on both the hole and electron doped sides. The Fermi surfaces near van Hove singularities admit partial nesting between valleys, which promotes both inter-valley supercon…
Due to their unique electron dispersion and lack of a Fermi surface, Coulomb interactions in undoped two-dimensional Dirac systems, such as single, bi- and tri-layer graphene, can be marginal or relevant. Relevant interactions can result in spontaneous symmetry breaking, which is responsible for a large class of physical phenomena ranging from mass generation in high energy physics to correlated states such as superconductivity and magnetism in condensed matter. Here, using transport measurements, we show that rhombohedral-stacked trilayer graphene (r-TLG) offers a simple, yet novel and tunabl…
Motivated by recent experimental observations of correlated metallic phases and superconductivity in rhombohedral trilayer graphene (RTG), we perform an unbiased study of electronic ordering instabilities in hole-doped RTG. Specifically, we focus on electronic states energetically proximate to Van Hove singularities (VHSs), where a large density of states promotes different interaction-induced symmetry-breaking electronic orders. To resolve the Fermi surface near VHSs, we construct a fermionic hot-spot model and demonstrate that a perpendicular electric field can tune different nesting structu…
We provide a complete theoretical interpretation of the metallic broken spin/valley symmetry states recently discovered in ABC trilayer graphene (ABC) perturbed by a large transverse displacement field. Our conclusions combine insights from ABC trilayer graphene electronic structure models and mean field theory, and are guided by precise magneto-oscillation Fermi-surface-area measurements. We conclude that the physics of ABC trilayer graphene is shaped by the principle of momentum-space condensation, which favors Fermi surface reconstructions enabled by broken spin/valley flavor symmetries whe…
Superconductivity was recently discovered in rhombohedral trilayer graphene (RTG) in the absence of a moiré potential. Superconductivity is observed proximate to a metallic state with reduced isospin symmetry, but it remains unknown whether this is a coincidence or a key ingredient for superconductivity. Using a Hartree-Fock analysis and constraints from experiments, we argue that the symmetry breaking is inter-valley coherent (IVC) in nature. We evaluate IVC fluctuations as a possible pairing glue, and find that they lead to chiral unconventional superconductivity when the fluctuations are st…
Study of the strongly correlated states in van der Waals heterostructures is one of the central topics in modern condensed matter physics. Among these, the rhombohedral trilayer graphene (RTG) occupies a prominent place since it hosts a variety of interaction-driven phases, with the metallic ones yielding exotic superconducting orders upon doping. Motivated by these experimental findings, we show within the framework of the low-energy Dirac theory that the optical conductivity can distinguish different candidates for a paramagnetic metallic ground state in this system. In particular, this obse…
Ferromagnetism is most common in transition metal compounds but may also arise in low-density two-dimensional electron systems, with signatures observed in silicon, III-V semiconductor systems, and graphene moiré heterostructures. Here we show that gate-tuned van Hove singularities in rhombohedral trilayer graphene drive the spontaneous ferromagnetic polarization of the electron system into one or more spin- and valley flavors. Using capacitance measurements on graphite-gated van der Waals heterostructures, we find a cascade of density- and electronic displacement field tuned phase transitions…
Multilayer graphene systems with a rhombohedral stacking order harbor nearly flat bands in their single-particle spectrum. We propose ansatz states to describe the surface-localized states of flat band electrons. The absence of kinetic dispersion near the fermi level leaves the interaction as a dominate mechanism to govern the low energy physics of a low density electron system. We build up an effective lattice model in two interacting low-energy bands, where the full terms of the Coulomb interaction, including those long-range and off-diagonal parts, have been considered. The interaction matr…
We report the observation of superconductivity in rhombohedral trilayer graphene electrostatically doped with holes. Superconductivity occurs in two distinct regions within the space of gate-tuned charge carrier density and applied electric displacement field, which we denote SC1 and SC2. The high sample quality allows for detailed mapping of the normal state Fermi surfaces by quantum oscillations, which reveal that in both cases superconductivity arises from a normal state described by an annular Fermi sea that is proximal to an isospin symmetry breaking transition where the Fermi surface deg…
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…
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…
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…
Recent transport experiments have demonstrated that the rhombohedral stacking trilayer graphene is an insulator with an intrinsic gap of 6meV and the Bernal stacking trilayer one is a metal. We propose a Hubbard model with a moderate $U$ for layered graphene sheets, and show that the model well explains the experiments of the stacking dependent energy gap. The on-site Coulomb repulsion drives the metallic phase of the non-interacting system to a weak surface antiferromagnetic insulator for the rhombohedral stacking layers, but does not alter the metallic phase for the Bernal stacking layers.
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…
Magnetic-field-induced superconductivity in hexalayer rhombohedral graphene
Laboratory of Atomic and Solid State Physics, Cornell University（康奈尔大学原子与固体物理实验室）；National Institute for Materials Science（国立材料研究所）；School of Applied and Engineering Physics, Cornell University（康奈尔大学应用工程物理学院）；Max Planck Institute for the Structure and Dynamics of Matter（马普物质结构与动力学研究所）
We use scanning tunneling microscopy to investigate Bi2Sr2Ca2Cu3O10+δ trilayer cuprates from the optimally doped to overdoped regime. We find that the two distinct superconducting gaps from the inner and outer CuO2 planes both decrease rapidly with doping, in sharp contrast to the nearly constant Tc. Spectroscopic imaging reveals the absence of quasiparticle interference in the antinodal region of overdoped samples, showing an opposite trend to that in single- and double-layer compounds. We propose that the existence of two types of inequivalent CuO2 planes and the intricate interaction betwee…
Surface-Selective Probe of Spin-Triplet Superconductivity in Rhombohedral Graphene
Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）；Department of Applied Physics, Aalto University School of Science（阿尔托大学理学院应用物理系）；Research Center for Functional Materials, National Institute for Materials Science（国立材料科学研究所功能材料研究中心）；International Center for Materials Nanoarchitectonics, National Institute for Materials Science（国立材料科学研究所国际纳米材料组装中心）；Braun Center for Submicron Research, Department of Condensed Matter Physics（雷霍沃特凝聚态物理系布朗亚微米研究中心）；Department of Physics, Florida State University（佛罗里达州立大学物理系）；National High Magnetic Field Laboratory（国家高磁场实验室）；Center for Quantum Science and Engineering (QSE Center), École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院量子科学与工程中心）
Based on a hybrid interlayer coupling mechanism, we study the coexistence of superconductivity (SC) and antiferromagnetism (AFM) in trilayer cuprates. By introducing an interlayer magnetic scattering term, we solve the multilayer $t{-}J$ model with Josephson coupling under the framework of Gutzwiller projection. We show that both the SC and AFM orders in the multilayered system are enhanced and the range of AFM order is extended. The layer configuration of d-wave pairing gap and AFM order further plays an essential role in determining the interlayer magnetic and superconducting coupling phase …
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…
Superconductivity and Ferroelectric Orbital Magnetism in Semimetallic Rhombohedral Hexalayer Graphene
Cornell University（康奈尔大学）；National Institute for Materials Science（物质材料研究机构）；School of Applied and Engineering Physics, Cornell University（康奈尔大学应用与工程物理学院）；Max Planck Institute for the Structure and Dynamics of Matter（马普物质结构与动力学研究所）
Incipient superconductivity and tunable Chern insulators in twisted bilayer-trilayer graphene
Department of Physics, University of Washington（华盛顿大学物理系）；Department of Materials Science and Engineering, University of Washington（华盛顿大学材料科学与工程系）；Research Center for Electronic and Optical Materials, National Institute for Materials Science（物质材料研究机构电子与光学材料研究中心）；Research Center for Materials Nanoarchitectonics, National Institute for Materials Science（物质材料研究机构的材料纳米结构研究中心）
Link between thermodynamic correlation signatures and superconductivity in twisted trilayer graphene
IBM Quantum（IBM量子）；Department of Physics, Stanford University（斯坦福大学物理系）；Geballe Laboratory for Advanced Materials（吉巴尔先进材料实验室）；Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory（斯坦福材料与能源科学研究所，SLAC国家加速器实验室）；Research Center for Electronic and Optical Materials, National Institute for Materials Science（物质材料研究机构电子与光学材料研究中心）；Research Center for Materials Nanoarchitectonics, National Institute for Materials Science（物质材料研究机构材料纳米架构中心）
The discovery of superconductivity in Sr-doped NdNiO$_{2}$ is a crucial breakthrough in the long pursuit for nickel oxide materials with electronic and magnetic properties similar to those of the cuprates. NdNiO$_2$ is the infinite-layer member of a family of square-planar nickelates with general chemical formula R$_{n+1}$Ni$_n$O$_{2n+2}$ (R = La, Pr, Nd, $n= 2, 3, ... \infty$). In this letter, we investigate superconductivity in the trilayer member of this series (R$_4$Ni$_3$O$_8$) using a combination of first-principles and $t-J$ model calculations. R$_4$Ni$_3$O$_8$ compounds resemble cuprat…
Mode-selective phonon effects on magnetism and superconductivity in trilayer nickelates
Universität Würzburg（维尔茨堡大学）；Nagoya University（名古屋大学）；Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所北京凝聚态物理国家研究中心）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Hefei National Laboratory, and New Cornerstone Science Laboratory（合肥国家实验室和新基石科学实验室）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）
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…
Quantum Geometry and Pair Breaking in Intra- and Intervalley Superconductivity of Rhombohedral Graphene
Theoretical Division, Los Alamos National Lab（洛斯阿拉莫斯国家实验室理论部）；Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA（田纳西大学诺克斯维尔分校物理与天文系）；Min H. Kao Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA（田纳西大学诺克斯维尔分校明浩高电气与计算机科学系）；Theoretical Division, T-4, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA（洛斯阿拉莫斯国家实验室T-4理论部）；Center for Integrated Nanotechnologies (CINT), Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA（洛斯阿拉莫斯国家实验室集成纳米技术中心）
Competing Orders Driven by Wigner Crystal Phase in Rhombohedral Graphene
Institute of Physics, Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院物理研究所）；SPEC, CEA, CNRS, Université Paris-Saclay（法国原子能和替代能源委员会、国家科学研究中心、萨克雷大学SPEC实验室）；Department of Physics, Florida State University（佛罗里达州立大学物理系）；National High Magnetic Field Laboratory（国家高磁场实验室）；FSU Quantum Initiative, Florida State University（佛罗里达州立大学FSU量子倡议项目）；Research Center for Functional Materials, National Institute for Materials Science（日本物质材料研究机构功能材料研究中心）
Motivated by the recently experimental reported signatures of the tunable Mott insulating state and superconductivity in an ABC graphene trilayer superlattice, we investigate the charge compressibility, the spin correlation, and the superconducting instability within the Hubbard model on a three-layer honeycomb lattice. It is found that an antiferromagnetically ordered Mott insulator emerges beyond a critical $U_c$ at half-filling, and the electronic correlation drives a $d+id$ superconducting pairing to be dominant over other pairing patterns in a wide doped region. The effective pairing inte…
A series of recent experiments have demonstrated robust superconductivity in magic-angle twisted trilayer graphene (TTG). In particular, a recent work by Cao et al. (arxiv:2103.12083) studies the behavior of the superconductor in an in-plane magnetic field and out-of-plane displacement field, finding that the superconductor is unlikely to be spin-singlet. This work also finds that at high magnetic fields and a smaller range of dopings and displacement fields, it undergoes a transition to a distinct field-induced superconducting state. Inspired by these results, we develop an understanding of s…
