Superlattice Potential Research Articles

Fractional Quantum Anomalous Hall Effect in Rhombohedral Multilayer Graphene in the Moiréless Limit.

The standard theoretical framework for fractional quantum anomalous Hall (FQAH) effect assumes an isolated flat Chern band in the single particle level. In this Letter, we challenge this paradigm for the FQAH effect recently observed in pentalayer rhombohedrally stacked graphene aligned with hexagonal boron nitride. We show that the external moiré superlattice potential is simply a perturbation in a model with continuous translation symmetry. Through Hartree-Fock calculations, we find that interaction opens a sizable remote-band gap, resulting in an isolated narrow C=1 Chern band at filling ν=1. From exact diagonalization we identify FQAH phases at various fillings. However, the FQAH states also exist in calculations without any external moiré potential. We suggest that the quantum anomalous Hall (QAH) insulator at ν=1 should be viewed as an interaction-driven topological Wigner crystal with QAH effect, which is subsequently pinned by a small moiré potential. The C=1 QAH crystal is robust with a crystal period around 10nm in 4-layer, 5-layer, 6-layer, and 7-layer graphene systems. Our work suggests a new direction to explore the interplay between topology and FQAH with spontaneous crystal formation in the vanishing moiré potential limit. We also propose a new system to generate and control both honeycomb and triangular moiré superlattice potentials through Coulomb interaction from another control layer, which can stabilize or suppress the QAH crystal depending on the density of the control layer.

Highly Tunable Moiré Superlattice Potentials in Twisted Hexagonal Boron Nitrides.

Moiré superlattice of twisted hexagonal boron nitride (hBN) has emerged as an advanced atomically thin van der Waals interfacial ferroelectricity platform. Nanoscale periodic ferroelectric moiré potentials in twisted hBN allow the hosting of remote Coulomb superlattice potentials to adjacent 2D materials. Therefore, the new strategies for engineering moiré length, angle, and potential strength are essential for developing programmable quantum materials. Here, it demonstrates the realization of twisted hBN-based moiré superlattice platforms and visualizes the moiré domains and ferroelectric properties using Kelvin probe force microscopy (KPFM). Also, the regular moiré superlattice in the large area is reported. It offers the possibility to reproduce uniform moiré structures with precise control piezo stage stacking and heat annealing. It demonstrates cumulative multi-ferroelectric polarization and multi-level domains with multiple angle mismatched interfaces. Additionally, it observes the quasi-1D anisotropic moiré domains and show the highest resolution analysis of the local built-in strain between adjacent hBN layers compared to the conventional methods. Furthermore, in-situ manipulation of moiré potential is demonstrated using femtosecond pulse laser, which results in the optical phonon-induced atomic displacement at the hBN moiré interfaces. The results pave the way to develop precisely programmable moiré superlattice platforms and investigate strongly correlated physics.

Signature of Correlated Insulator in Electric Field Controlled Superlattice.

On a two-dimensional crystal, a "superlattice" with nanometer-scale periodicity can be imposed to tune the Bloch electron spectrum, enabling novel physical properties inaccessible in the original crystal. While creating 2D superlattices by means of nanopatterned electric gates has been studied for band structure engineering in recent years, evidence of electron correlations─which drive many problems at the forefront of physics research─remains to be uncovered. In this work, we demonstrate signatures of a correlated insulator phase in Bernal-stacked bilayer graphene modulated by a gate-defined superlattice potential, manifested as resistance peaks centered at integer multiples of single electron per superlattice unit cell carrier densities. The observation is consistent with the formation of a stack of flat low-energy bands due to the superlattice potential combined with inversion symmetry breaking. Our work paves the way to custom-designed superlattices for studying band structure engineering and strongly correlated electrons in 2D materials.

Understanding disorder in monolayer graphene devices with gate-defined superlattices

Engineering superlattices (SLs)-which are spatially periodic potential landscapes for electrons-is an emerging approach for the realization of exotic properties, including superconductivity and correlated insulators, in two-dimensional materials. While moiré SL engineering has been a popular approach, nanopatterning is an attractive alternative offering control over the pattern and wavelength of the SL. However, the disorder arising in the system due to imperfect nanopatterning is seldom studied. Here, by creating a square lattice of nanoholes in the SiO2dielectric layer using nanolithography, we study the SL potential and the disorder formed in hBN-graphene-hBN heterostructures. Specifically, we observe that while electrical transport shows distinct SL satellite peaks, the disorder of the device is significantly higher than graphene devices without any SL. We use finite-element simulations combined with a resistor network model to calculate the effects of this disorder on the transport properties of graphene. We consider three types of disorder: nanohole size variations, adjacent nanohole mergers, and nanohole vacancies. Comparing our experimental results with the model, we find that the disorder primarily originates from nanohole size variations rather than nanohole mergers in square SLs. We further confirm the validity of our model by comparing the results with quantum transport simulations. Our findings highlight the applicability of our simple framework to predict and engineer disorder in patterned SLs, specifically correlating variations in the resultant SL patterns to the observed disorder. Our combined experimental and theoretical results could serve as a valuable guide for optimizing nanofabrication processes to engineer disorder in nanopatterned SLs.

Polarization textures in crystal supercells with topological bands

Two-dimensional materials are a highly tunable platform for studying the momentum space topology of the electronic wavefunctions and real space topology in terms of skyrmions, merons, and vortices of an order parameter. Such textures for electronic polarization can exist in moiré heterostructures. A quantum-mechanical definition of local polarization textures in insulating supercells was recently proposed. Here, we propose a definition for local polarization that is also valid for systems with topologically nontrivial bands. We introduce semilocal hybrid polarizations, which are valid even when the Wannier functions in a system cannot be made exponentially localized in all dimensions. We use this definition to explicitly show that nontrivial real-space polarization textures can exist in topologically nontrivial systems with nonzero Chern number under (1) an external superlattice potential, and (2) under a stacking-induced moiré potential. In the latter, we find that while the magnitude of the local polarization decreases discontinuously across a topological phase transition from trivial to topologically nontrivial, the polarization does not completely vanish. Our findings suggest that band topology and real-space polar topology may coexist in real materials. Published by the American Physical Society 2024

Modulation of Electrostatic Potential in 2D Crystal Engineered by an Array of Alternating Polar Molecules.

The moiré potential in rotationally misfit two-dimensional (2D) heterostructures has been used to build artificial exciton and electron lattices, which have become platforms for realizing exotic electronic phases. Here, we demonstrate a different approach to create a superlattice potential in 2D crystals by using the near field of an array of polar molecules. A bilayer of titanyl phthalocyanine (TiOPc), consisting of alternating out-of-plane dipoles, is deposited on monolayer MoS2. Time-resolved two-photon photoemission spectroscopy reveals a pair of interlayer exciton states with an energy difference of ∼0.1 eV, which is consistent with the electrostatic potential modulation induced by the TiOPc bilayer as determined by density functional theory calculations. Because the symmetry and the period of this potential superlattice can be changed readily by using molecules of different shapes and sizes, molecule/2D heterostructures can be promising platforms for designing artificial exciton and electron lattices.

Interplay of valley, layer and band topology towards interacting quantum phases in moiré bilayer graphene

In Bernal-stacked bilayer graphene (BBG), the Landau levels give rise to an intimate connection between valley and layer degrees of freedom. Adding a moiré superlattice potential enriches the BBG physics with the formation of topological minibands — potentially leading to tunable exotic quantum transport. Here, we present magnetotransport measurements of a high-quality bilayer graphene–hexagonal boron nitride (hBN) heterostructure. The zero-degree alignment generates a strong moiré superlattice potential for the electrons in BBG and the resulting Landau fan diagram of longitudinal and Hall resistance displays a Hofstadter butterfly pattern with a high level of detail. We demonstrate that the intricate relationship between valley and layer degrees of freedom controls the topology of moiré-induced bands, significantly influencing the energetics of interacting quantum phases in the BBG superlattice. We further observe signatures of field-induced correlated insulators, helical edge states and clear quantizations of interaction-driven topological quantum phases, such as symmetry broken Chern insulators.

A Tunable Graphene Superlattice with Deformable Periodical Nano-Gating.

Graphene superlattices have simple and controllable electronic band structures, which can also be electrostatically tuned. They have been widely studied for band engineering and strong correlated physics, and have led to the discovery of a variety of exciting phenomena. To experimentally study the physics of graphene superlattices in a systematic way, it is desirable to control the structure parameters, which barely exist at the moment, onsite. Here, a tunable superlattice with graphene and a deformable gating structure is demonstrated. The period and duty cycle of the nano-gating, and furthermore of the superlattice potential, can be tuned through altering the shape of the gating structure with piezo-actuators, offering a tunable band structure. The tuning of the electronic band structures of both a two-dimensional and a one-dimensional superlattice is demonstrated with numerical simulations, offering a new approach for tunable electronic and photonic devices.

Topological superconductivity induced by spin-orbit coupling, perpendicular magnetic field, and superlattice potential

Topological superconductivity induced by spin-orbit coupling, perpendicular magnetic field, and superlattice potential

Tunable magnetic confinement effect in a magnetic superlattice of graphene

Two-dimensional van der Waals materials such as graphene present an opportunity for band structure engineering using custom superlattice potentials. In this study, we demonstrate how self-assemblies of magnetic iron-oxide (Fe3O4) nanospheres stacked on monolayer graphene generate a proximity-induced magnetic superlattice in graphene and modify its band structure. Interactions between the nanospheres and the graphene layer generate superlattice Dirac points in addition to a gapped energy spectrum near the K and K′ valleys, resulting in magnetic confinement of quasiparticles around the nanospheres. This is evidenced by gate-dependent resistance oscillations, observed in our low temperature transport measurements, and confirmed by self-consistent tight binding calculations. Furthermore, we show that an external magnetic field can tune the magnetic superlattice potential created by the nanospheres, and thus the transport characteristics of the system. This technique for magnetic-field-tuned band structure engineering using magnetic nanostructures can be extended to a broader class of 2D van der Waals and topological materials.

Topological minibands and interaction driven quantum anomalous Hall state in topological insulator based moiré heterostructures

The presence of topological flat minibands in moiré materials provides an opportunity to explore the interplay between topology and correlation. In this work, we study moiré minibands in topological insulator films with two hybridized surface states under a moiré superlattice potential created by two-dimensional insulating materials. We show the lowest conduction (highest valence) Kramers’ pair of minibands can be Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} non-trivial when the minima (maxima) of moiré potential approximately form a hexagonal lattice with six-fold rotation symmetry. Coulomb interaction can drive the non-trivial Kramers’ minibands into the quantum anomalous Hall state when they are half-filled, which is further stabilized by applying external gate voltages to break inversion. We propose the monolayer Sb2 on top of Sb2Te3 films as a candidate based on first principles calculations. Our work demonstrates the topological insulator based moiré heterostructure as a potential platform for studying interacting topological phases.

Nonlinear Landau Fan Diagram for Graphene Electrons Exposed to a Moiré Potential.

Due to Landau quantization, the conductance of two-dimensional electrons exposed to a perpendicular magnetic field exhibits oscillations that generate a fan of linear trajectories when plotted in the parameter space spanned by density and field. This fan looks identical, irrespective of the dispersion and field dependence of the Landau level energy. This is no surprise because the position of conductance minima depends solely on the level degeneracy that is linear in flux. The fractal energy spectrum that emerges within each Landau band when electrons are also exposed to a two-dimensional superlattice potential produces numerous additional oscillations, but they also create just linear fans for identical reasons. Here, we report conductance oscillations of graphene electrons exposed to a moiré potential that defy this general rule and form nonlinear trajectories in the density-field plane. We attribute this anomalous behavior to the simultaneous occupation of multiple minibands and magnetic breakdown-induced open orbits.

Dynamically tunable moiré exciton Rydberg states in a monolayer semiconductor on twisted bilayer graphene.

Moiré excitons are emergent optical excitations in two-dimensional semiconductors with moiré superlattice potentials. Although these excitations have been observed on several platforms, a system with dynamically tunable moiré potential to tailor their properties is yet to be realized. Here we present a continuously tunable moiré potential in monolayer WSe2, enabled by its proximity to twisted bilayer graphene (TBG) near the magic angle. By tuning local charge density via gating, TBG provides a spatially varying and dynamically tunable dielectric superlattice for modulation of monolayer WSe2 exciton wave functions. We observed emergent moiré exciton Rydberg branches with increased energy splitting following doping of TBG due to exciton wave function hybridization between bright and dark Rydberg states. In addition, emergent Rydberg states can probe strongly correlated states in TBG at the magic angle. Our study provides a new platform for engineering moiré excitons and optical accessibility to electronic states with small correlation gaps in TBG.

Conductance and polarization in an 8-Pmmn borophene-based superlattice

We theoretically investigate the effect of the Rashba interaction on the conductance as well as the valley and spin polarization in an 8-Pmmn borophene superlattice by solving the Dirac–like equation and using the transfer matrix approach. The valley and spin-dependent conductance and spin polarization are effectively controlled by changing the direction of superlattice. The number of the quantum electrostatic potential plays an essential role in quantum transmission properties. Furthermore, in such a superlattice, the conductance of the electrons with spin rotation can be easily adjusted by the Rashba strength, direction of superlattice and electrostatic potential. Thus, borophene superlattice with Rashba interaction can act as a spin and valley splitter. Our work provides a new structure for spin-flip and can be useful for future spintronics and valleytronics applications.

Observation of a Folded Dirac Cone in Heavily Doped Graphene.

Superlattice potentials imposed on graphene can alter its Dirac states, enabling the realization of various quantum phases. We report the experimental observation of a replica Dirac cone at the Brillouin zone center induced by a superlattice in heavily doped graphene with Gd intercalation using angle-resolved photoemission spectroscopy (ARPES). The replica Dirac cone arises from the (√3× √3)R30° superlattice formed by the intervalley coupling of two nonequivalent valleys (e.g., the Kekulé-like distortion phase), accompanied by a bandgap opening. According to the findings, the replica Dirac band in Gd-intercalated graphene disappears beyond a critical temperature of 30 K and can be suppressed by potassium adsorption. The modulation of the replica Dirac band is primarily attributable to the residual frozen gas, which can act as a source of intervalley scattering at temperatures below 30 K. Our results highlight the persistence of the hidden Kekulé-like phase within the heavily doped graphene, enriching our current understanding of its replica Dirac Fermions.

Majorana gap formation in the anisotropic Kitaev model with ordered flux configuration

We study the Kitaev model with direction dependent interactions to investigate how the flux configuration and/or the anisotropy in the exchanges affect the Majorana excitations. Systematic numerical calculations demonstrate how the anisotropy of the exchange couplings and flux configuration make the Majorana excitation gapped. The induced gapped quantum spin liquid states are distinct from the gapped one realized in the large anisotropic limit. The nature of gapped states can be explained by the superlattice potential due to flux configuration.

Multilayer graphene with a superlattice potential

Bernal stacked bilayer graphene subject to a superlattice potential can realize topological and stacked flat bands [Ghorashi et al., Phys. Rev. Lett. 130, 196201 (2023)]. In the present work, we extend the study of a superlattice potential on graphene heterostructures to trilayer and quadrilayer graphene. Comparing Bernal- and chirally stacked multilayers reveals that the latter are more suitable for realizing stacks of many flat bands. On the other hand, Bernal-stacked graphene heterostructures can realize topological flat bands. Imposing two simultaneous superlattice potentials enhances the viability of both regimes.

Topological and Stacked Flat Bands in Bilayer Graphene with a Superlattice Potential.

We show that bilayer graphene in the presence of a 2D superlattice potential provides a highly tunable setup that can realize a variety of flat band phenomena. We focus on two regimes: (i)topological flat bands with nonzero Chern numbers, C, including bands with higher Chern numbers |C|>1 and (ii)an unprecedented phase consisting of a stack of nearly perfect flat bands with C=0. For realistic values of the potential and superlattice periodicity, this stack can span nearly 100meV, encompassing nearly all of the low-energy spectrum. We further show that in the topological regime, the topological flat band has a favorable band geometry for realizing a fractional Chern insulator (FCI) and use exact diagonalization to show that the FCI is in fact the ground state at 1/3 filling. Our results provide a realistic guide for future experiments to realize a new platform for flat band phenomena.

Emergence of tunable intersubband-plasmon-polaritons in graphene superlattices

On-demand modification of the electronic band structures of high-mobility two-dimensional (2D) materials is of great interest for various applications that require rapid tuning of electrical and optical responses of solid-state devices. Although electrically tunable superlattice (SL) potentials have been proposed for band structure engineering of the Dirac electrons in graphene, the ultimate goal of engineering emergent quasiparticle excitations that can hybridize with light has not been achieved. We show that an extreme modulation of one-dimensional (1D) SL potentials in monolayer graphene produces ladder-like electronic energy levels near the Fermi surface, resulting in optical conductivity dominated by intersubband transitions (ISBTs). A specific and experimentally realizable platform comprising hBN-encapsulated graphene on top of a 1D periodic metagate and a second unpatterned gate is shown to produce strongly modulated electrostatic potentials. We find that Dirac electrons with large momenta perpendicular to the modulation direction are waveguided via total internal reflections off the electrostatic potential, resulting in flat subbands with nearly equispaced energy levels. The predicted ultrastrong coupling of surface plasmons to electrically controlled ISBTs is responsible for emergent polaritonic quasiparticles that can be optically probed. Our study opens an avenue for exploring emergent polaritons in 2D materials with gate-tunable electronic band structures.

Chiral Decomposition of Twisted Graphene Multilayers with Arbitrary Stacking.

We formulate the chiral decomposition rules that govern the electronic structure of a broad family of twisted N + M multilayer graphene configurations that combine arbitrary stacking order and a mutual twist. We show that at the magic angle in the chiral limit the low-energy bands of such systems are composed of chiral pseudospin doublets that are energetically entangled with two flat bands per valley induced by the moiré superlattice potential. The analytic construction is supported by explicit numerical calculations based on realistic parametrization. We further show that vertical displacement fields can open energy gaps between the pseudospin doublets and the two flat bands, such that the flat bands may carry nonzero valley Chern numbers. These results provide guidelines for the rational design of topological and correlated states in generic twisted graphene multilayers.