Making massless Dirac fermions from a patterned two-dimensional electron gas

A Novel Quasi-Planar Two-dimensional Carbon Sulfide with Negative Poisson's Ratio and Dirac Fermions

In the present study, a novel and unconventional two-dimensional (2D) material with Dirac electronic features has been designed using sulflower with the help of density functional theory methods and first principles calculations. This 2D material comprises of hetero atoms (C, S) and belongs to the tetragonal lattice with P4 /nmm space group. Scrutiny of the results show that the 2D nanosheet exhibits a nanoporous wave-like geometrical structure. Quantum molecular dynamics simulations and phonon mode analysis emphasize the dynamical and thermal stability. The novel 2D nanosheet is an auxetic material with an anisotropy in the in-plane mechanical properties. Both composition and geometrical features are completely different from the conditions necessary for the formation of Dirac cones in graphene. However, the presence of semi-metallic nature, linear band dispersion relation, massive fermions and massless Dirac fermions are observed in the novel 2D nanosheet. The massless Dirac fermions exhibit highly isotropic Fermi velocities (vf =0.68×106  m/s) along all crystallographic directions. The zero-band gap semi metallic features of the novel 2D nanosheet are perturbative to the electric field and external strain.

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 100 meV, 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.

Formation of Artificial Fermi Surfaces with a Triangular Superlattice on a Conventional Two-Dimensional Electron Gas

Imposing an external periodic electrostatic potential to the electrons confined in a quantum well makes it possible to engineer synthetic two-dimensional band structures, with electronic properties different from those in the host semiconductor. Here we report the fabrication and study of a tunable triangular artificial lattice on a GaAs/AlGaAs heterostructure where it is possible to transform from the original GaAs band structure and a circular Fermi surface to a new band structure with multiple artificial Fermi surfaces simply by altering a gate bias. For weak electrostatic modulation magnetotransport measurements reveal multiple quantum oscillations and commensurability oscillations due to the electron scattering from the artificial lattice. Increasing the strength of the modulation reveals new commensurability oscillations of the electrons from the artificial Fermi surface scattering from the triangular artificial lattice. These results show that low disorder gate-tunable lateral superlattices can be used to form artificial two-dimensional crystals with designer electronic properties.

Tunable Mott Dirac and Kagome Bands Engineered on 1T-TaS2

Strongly interacting electrons in hexagonal and kagome lattices exhibit rich phase diagrams of exotic quantum states, including superconductivity and correlated topological orders. However, material realizations of these electronic states have been scarce in nature or by design. Here, we theoretically propose an approach to realize artificial lattices by metal adsorption on a 2D Mott insulator 1T-TaS₂. Alkali, alkaline-earth, and group 13 metal atoms are deposited in (√3 × √3)R30° and 2 × 2 TaS₂ superstructures of honeycomb- and kagome-lattice symmetries exhibiting Dirac and kagome bands, respectively. The strong electron correlation of 1T-TaS₂ drives the honeycomb and kagome systems into correlated topological phases described by Kane-Mele-Hubbard and kagome-Hubbard models. We further show that the 2/3 or 3/4 band filling of Mott Dirac and flat bands can be achieved with a proper concentration of Mg adsorbates. Our proposal may be readily implemented in experiments, offering an attractive condensed-matter platform to exploit the interplay of correlated topological order and superconductivity.

On the problem of Dirac cones in fullerenes on gold

Artificial graphene based on molecular networks enables the creation of novel 2D materials with unique electronic and topological properties. Landau quantization has been demonstrated by CO molecules arranged on the two-dimensional electron gas on Cu(111) and the observation of electron quantization may succeed based on the created gauge fields. Recently, it was reported that instead of individual manipulation of CO molecules, simple deposition of nonpolar C₆₀ molecules on Cu(111) and Au(111) produces artificial graphene as evidenced by Dirac cones in photoemission spectroscopy. Here, we show that C₆₀-induced Dirac cones on Au(111) have a different origin. We argue that those are related to umklapp diffraction of surface electronic bands of Au on the molecular grid of C₆₀ in the final state of photoemission. We test this alternative explanation by precisely probing the dimensionality of the observed conical features in the photoemission spectra, by varying both the incident photon energy and the degree of charge doping via alkali adatoms. Using density functional theory calculations and spin-resolved photoemission we reveal the origin of the replicating Au(111) bands and resolve them as deep leaky surface resonances derived from the bulk Au sp-band residing at the boundary of its surface projection. We also discuss the manifold nature of these resonances which gives rise to an onion-like Fermi surface of Au(111).

Electronic Quantum Materials Simulated with Artificial Model Lattices

The band structure and electronic properties of a material are defined by the sort of elements, the atomic registry in the crystal, the dimensions, the presence of spin–orbit coupling, and the electronic interactions. In natural crystals, the interplay of these factors is difficult to unravel, since it is usually not possible to vary one of these factors in an independent way, keeping the others constant. In other words, a complete understanding of complex electronic materials remains challenging to date. The geometry of two- and one-dimensional crystals can be mimicked in artificial lattices. Moreover, geometries that do not exist in nature can be created for the sake of further insight. Such engineered artificial lattices can be better controlled and fine-tuned than natural crystals. This makes it easier to vary the lattice geometry, dimensions, spin–orbit coupling, and interactions independently from each other. Thus, engineering and characterization of artificial lattices can provide unique insights. In this Review, we focus on artificial lattices that are built atom-by-atom on atomically flat metals, using atomic manipulation in a scanning tunneling microscope. Cryogenic scanning tunneling microscopy allows for consecutive creation, microscopic characterization, and band-structure analysis by tunneling spectroscopy, amounting in the analogue quantum simulation of a given lattice type. We first review the physical elements of this method. We then discuss the creation and characterization of artificial atoms and molecules. For the lattices, we review works on honeycomb and Lieb lattices and lattices that result in crystalline topological insulators, such as the Kekulé and “breathing” kagome lattice. Geometric but nonperiodic structures such as electronic quasi-crystals and fractals are discussed as well. Finally, we consider the option to transfer the knowledge gained back to real materials, engineered by geometric patterning of semiconductor quantum wells.

Engineering novel surface electronic states via complex supramolecular tessellations

Tailoring Shockley surface-state (SS) electrons utilizing complex interfacial supramolecular tessellations was explored by low-temperature scanning tunnelling microscopy and spectroscopy, combined with computational modelling using electron plane wave expansion (EPWE) and empirical tight-binding (TB) methods. Employing a recently introduced gas-mediated on-surface reaction protocol, three distinct types of open porous networks comprising paired organometallic species as basic tectons were selectively synthesized. In particular, these supramolecular networks feature semiregular Archimedean tilings, providing intricate quantum dots (QDs) coupling scenarios compared to hexagonal porous superlattices. Our experimental results in conjunction with modelling calculations demonstrate the possibility of realizing novel two-dimensional electronic structures such as Kagome- and Dirac-type as well as hybrid Kagome-type bands via QD coupling. Compared to constructing SS electron pathways via molecular manipulations, our studies reveal significant potential of exploiting QD coupling as a complementary and versatile route for the control of surface electronic landscapes.

Artificial Graphene on Si Substrates: Fabrication and Transport Characteristics

Artificial graphene (AG) based on a honeycomb lattice of semiconductor quantum dots (QDs) has been of great interest for exploration and applications of massless Dirac Fermions in semiconductors thanks to the tunable interplay between the carrier interactions and the honeycomb topology. Here, an innovative strategy to realize AG on Si substrates is developed by fabricating a honeycomb lattice of Au nanodisks on a Si/GeSi quantum well. The lateral potential modulation induced by the nanoscale Au/Si Schottky junction results in the formation of quantum dots arranged in a honeycomb lattice to form AG. Nonlinear current-voltage curves of the AG reveal conductance phase transitions with switch on/off voltages, a large electric hysteresis loop, and a strong sharp current peak accompanied by a group of differential-conductance peaks and negative differential conductance around the switch-on voltage, which can be modulated by temperature and light. These features are interpreted by a model based on the Coulomb blockade effect, the collective resonant tunneling, and the coupling of holes in the AG. Our results not only demonstrate an approach to the formation but also will greatly stimulate the characterizations and the applications of innovative semiconductor-based AG.

Electronic properties of atomically coherent square PbSe nanocrystal superlattice resolved by Scanning Tunneling Spectroscopy

Rock-salt lead selenide nanocrystals can be used as building blocks for large scale square superlattices via two-dimensional assembly of nanocrystals at a liquid-air interface followed by oriented attachment. Here we report Scanning Tunneling Spectroscopy measurements of the local density of states of an atomically coherent superlattice with square geometry made from PbSe nanocrystals. Controlled annealing of the sample permits the imaging of a clean structure and to reproducibly probe the band gap and the valence hole and conduction electron states. The measured band gap and peak positions are compared to the results of optical spectroscopy and atomistic tight-binding calculations of the square superlattice band structure. In spite of the crystalline connections between nanocrystals that induce significant electronic couplings, the electronic structure of the superlattices remains very strongly influenced by the effects of disorder and variability.

Investigation of edge states in artificial graphene nano-flakes

Graphene nano-flakes (GNFs) are predicted to host spin-polarized metallic edge states, which are envisioned for exploration of spintronics at the nanometer scale. To date, experimental realization of GNFs is only in its infancy because of the limitation of precise cutting or synthesizing methods at the nanometer scale. Here, we use low temperature scanning tunneling microscope to manipulate coronene molecules on a Cu(111) surface to build artificial triangular and hexagonal GNFs with either zigzag or armchair type of edges. We observe that an electronic state at the Dirac point emerges only in the GNFs with zigzag edges and localizes at the outmost lattice sites. The experimental results agree well with the tight-binding calculations. Our work renders an experimental confirmation of the predicated edge states of the GNFs.

Observation of Flat Bands in Gated Semiconductor Artificial Graphene

Flat bands near M points in the Brillouin zone are key features of honeycomb symmetry in artificial graphene (AG) where electrons may condense into novel correlated phases. Here we report the observation of van Hove singularity doublet of AG in GaAs quantum well transistors, which presents the evidence of flat bands in semiconductor AG. Two emerging peaks in photoluminescence spectra tuned by backgate voltages probe the singularity doublet of AG flat bands and demonstrate their accessibility to the Fermi level. As the Fermi level crosses the doublet, the spectra display dramatic stability against electron density, indicating interplays between electron-electron interactions and honeycomb symmetry. Our results provide a new flexible platform to explore intriguing flat band physics.

The realization of quantum anomalous Hall effect in two dimensional electron gas

The quantum anomalous Hall effect (QAHE), carrying dissipationless chiral edge states, occurs without any magnetic field. Two main strategies were proposed to host QAHE: the magnetic topological insulator thin films and graphene systems. Only the former one was realized in experiment at low temperature. In this paper, by dealing with the two-dimensional electron gas with an anti-dot lattice, a realistic platform is proposed to host the QAHE with both Chern number [Formula: see text] and [Formula: see text]. Based on the calculation of the Berry curvature integral and spacial wave function, the topological nature of the QAH edge states is well demonstrated. In the QAH region, the conductance shows quantized plateaus and their values are robust against Anderson disorder. In addition, we have also studied the effects of the size and shape of the anti-dot lattice on QAHE and they provide extra manners to adjust the system parameters. Taking the advantages of the well developed micro-manufacture technique in semiconductors, the proposal is experimentally accessible in micro-scale.

Direct observation of Klein tunneling in phononic crystals

Tunneling plays an essential role in many branches of physics and has found important applications. It is theoretically proposed that Klein tunneling occurs when, under normal incidence, quasiparticles exhibit unimpeded penetration through potential barriers independent of their height and width. We created a phononic heterojunction by sandwiching two types of artificial phononic crystals with different Dirac point energies. The direct observation of Klein tunneling as shown by the key feature of unity transmission is demonstrated. Our experiment reveals that Klein tunneling occurs over a broad band of acoustic frequency. The direct observation of Klein tunneling in phononic crystals could find applications in signal processing, supercollimated beams, and communications.

Mapping Defect Relaxation in Quantum Dot Solids upon In Situ Heating

Epitaxially connected quantum dot solids have emerged as an interesting class of quantum confined materials with the potential for highly tunable electronic structures. Realization of the predicted emergent electronic properties has remained elusive due in part to defective interdot epitaxial connections. Thermal annealing has shown potential to eliminate such defects, but a direct understanding of this mechanism hinges on determining the nature of defects in the connections and how they respond to heating. Here, we use in situ heating in the scanning transmission electron microscope to probe the effect of heating on distinct defect types. We apply a real space, local strain mapping technique, which allows us to identify tensile and shear strain in the atomic lattice, highlighting tensile, shear, and bending defects in interdot connections. We also track the out-of-plane orientation of individual QDs and infer the prevalence of out-of-plane twisting and bending defects as a function of annealing. We find that tensile and shear defects are fully relaxed upon mild thermal annealing, while bending defects persist. Additionally, out-of-plane orientation tracking reveals an increase in correctly oriented QDs, pointing to a relaxation of either twisting defects or out-of-plane bending defects. While bending defects remain, highlighting the need for further study of orientational ordering during the preattachment phase of superlattice formation, these atomic-scale insights show that annealing can effectively eliminate tensile and shear defects, a promising step toward delocalization of charge carriers and tunable electronic properties.

Ultrahigh-yield on-surface synthesis and assembly of circumcoronene into a chiral electronic Kagome-honeycomb lattice

On-surface synthesis has revealed remarkable potential in the fabrication of atomically precise nanographenes. However, surface-assisted synthesis often involves multiple-step cascade reactions with competing pathways, leading to a limited yield of target nanographene products. Here, we devise a strategy for the ultrahigh-yield synthesis of circumcoronene molecules on Cu(111) via surface-assisted intramolecular dehydrogenation of the rationally designed precursor, followed by methyl radical-radical coupling and aromatization. An elegant electrostatic interaction between circumcoronenes and metallic surface drives their self-organization into an extended superlattice, as revealed by bond-resolved scanning probe microscopy measurements. Density functional theory and tight-binding calculations reveal that unique hexagonal zigzag topology of circumcoronenes, along with their periodic electrostatic landscape, confines two-dimensional electron gas in Cu(111) into a chiral electronic Kagome-honeycomb lattice with two emergent electronic flat bands. Our findings open up a new route for the high-yield fabrication of elusive nanographenes with zigzag topologies and their superlattices with possible nontrivial electronic properties.

p Orbital Flat Band and Dirac Cone in the Electronic Honeycomb Lattice

Theory anticipates that the in-plane px, py orbitals in a honeycomb lattice lead to potentially useful quantum electronic phases. So far, p orbital bands were only realized for cold atoms in optical lattices and for light and exciton-polaritons in photonic crystals. For electrons, in-plane p orbital physics is difficult to access since natural electronic honeycomb lattices, such as graphene and silicene, show strong s-p hybridization. Here, we report on electronic honeycomb lattices prepared on a Cu(111) surface in a scanning tunneling microscope that, by design, show (nearly) pure orbital bands, including the p orbital flat band and Dirac cone.

Resilient Pathways to Atomic Attachment of Quantum Dot Dimers and Artificial Solids from Faceted CdSe Quantum Dot Building Blocks

The goal of this work is to identify favored pathways for preparation of defect-resilient attached wurtzite CdX (X = S, Se, Te) nanocrystals. We seek guidelines for oriented attachment of faceted nanocrystals that are most likely to yield pairs of nanocrystals with either few or no electronic defects or electronic defects that are in and of themselves desirable and stable. Using a combination of in situ high-resolution transmission electron microscopy (HRTEM) and electronic structure calculations, we evaluate the relative merits of atomic attachment of wurtzite CdSe nanocrystals on the {11̅00} or {112̅0} family of facets. Pairwise attachment on either facet can lead to perfect interfaces, provided the nanocrystal facets are perfectly flat and the angles between the nanocrystals can adjust during the assembly. Considering defective attachment, we observe for {11̅00} facet attachment that only one type of edge dislocation forms, creating deep hole traps. For {112̅0} facet attachment, we observe that four distinct types of extended defects form, some of which lead to deep hole traps whereas others only to shallow hole traps. HRTEM movies of the dislocation dynamics show that dislocations at {11̅00} interfaces can be removed, albeit slowly. Whereas only some extended defects at {112̅0} interfaces could be removed, others were trapped at the interface. Based on these insights, we identify the most resilient pathways to atomic attachment of pairs of wurtzite CdX nanocrystals and consider how these insights can translate to the creation of electronically useful materials from quantum dots with other crystal structures.

Flat AgTe Honeycomb Monolayer on Ag(111)

The intriguing properties of graphene have inspired the pursuit of two-dimensional materials with honeycomb structure. Here we achieved the synthesis of a monolayer transition-metal monochalcogenide AgTe on Ag(111) by tellurization of the substrate. High-resolution scanning tunneling microscopy, combined with low-energy electron diffraction, angle-resolved photoemission spectroscopy, and density functional theory calculations, demonstrates the planar honeycomb structure of AgTe. The first-principles calculations further predict that, protected by the in-plane mirror reflection symmetry, there are two Dirac node-line fermions existing in the free-standing AgTe when spin-orbit coupling (SOC) is ignored. In fact, the SOC leads to the gap opening, resulting in the emergence of the topologically nontrivial quantum spin Hall edge state. Importantly, our experiments evidence the chemical stability of the monolayer AgTe in ambient conditions, making it possible to study AgTe by more ex situ measurements and even to utilize AgTe in future electronic devices.

Triangular nanoperforation and band engineering of InGaAs quantum wells: a lithographic route toward Dirac cones in III-V semiconductors

The design of two-dimensional periodic structures at the nanoscale has renewed attention for band structure engineering. Here, we investigate the nanoperforation of InGaAs quantum wells epitaxially grown on InP substrates using high-resolution e-beam lithography and highly plasma based dry etching. We report on the fabrication of a honeycomb structure with an effective lattice constant down to 23 nm by realising triangular antidot lattice with an ultimate periodicity of 40 nm in a 10 nm thick InGaAs quantum well on a p-type InP. The quality of the honeycomb structures is discussed in detail, and calculations show the possibility to measure Dirac physics in these type of samples. Based on the statistical analysis of the fluctuations in pore size and periodicity, calculations of the band structure are performed to assess the robustness of the Dirac cones with respect to distortions of the honeycomb lattice.

New topological states in HgTe quantum wells from defect patterning

To explore new methods for the realization of the quantum spin Hall (QSH) effect in two-dimensional (2D) materials, we have constructed a honeycomb geometry (HG) by etching rows of hexagonal holes in HgTe quantum wells (QWs). Theoretical calculations show that multiple Dirac cones can be produced by HG, regardless of whether the band inversion occurs or not. Furthermore, the topological states originating from a narrow HG region in a wide ribbon show strong localization at the physical edges of the ribbon, making them easy to manipulate and exploit. When the band inversion condition for QW states is satisfied, the topological states generated by two different mechanisms may coexist. Our studies pave the way to produce and control multiple QSH states in 2D materials as desired for the design of innovative spintronic materials.

Emerging many-body effects in semiconductor artificial graphene with low disorder

The interplay between electron-electron interactions and the honeycomb topology is expected to produce exotic quantum phenomena and find applications in advanced devices. Semiconductor-based artificial graphene (AG) is an ideal system for these studies that combines high-mobility electron gases with AG topology. However, to date, low-disorder conditions that reveal the interplay of electron-electron interaction with AG symmetry have not been achieved. Here, we report the creation of low-disorder AG that preserves the near-perfection of the pristine electron layer by fabricating small period triangular antidot lattices on high-quality quantum wells. Resonant inelastic light scattering spectra show collective spin-exciton modes at the M-point's nearly flatband saddle-point singularity in the density of states. The observed Coulomb exchange interaction energies are comparable to the gap of Dirac bands at the M-point, demonstrating interplay between quasiparticle interactions and the AG potential. The saddle-point exciton energies are in the terahertz range, making low-disorder AG suitable for contemporary optoelectronic applications.

Topological States Characterized by Mirror Winding Numbers in Graphene with Bond Modulation

Localized electrons appear at the zigzag-shaped edge of graphene due to quantum interference. Here we propose a way for harnessing the edge electronic states to make them mobile, by incorporating a topological view point. The manipulation required is to introduce a pattern of strong-weak bonds between neighboring carbon atoms, and to put side by side two graphene sheets with strong-weak alternation conjugating to each other. The electrons with up and down pseudospins propagate in opposite directions at the interface, similar to the prominent quantum spin Hall effect. The system is characterized by a topological index, the mirror winding number, with its root lying in the Su-Schrieffer-Heeger model for polymer. Taking this point of view, one is rewarded by several ways for decorating graphene edge which result in similar mobile electronic states with topological protection. This work demonstrates that celebrated nanotechnology can be used to derive topological states.

Experimental realization and characterization of an electronic Lieb lattice

Geometry, whether on the atomic or nanoscale, is a key factor for the electronic band structure of materials. Some specific geometries give rise to novel and potentially useful electronic bands. For example, a honeycomb lattice leads to Dirac-type bands where the charge carriers behave as massless particles [1]. Theoretical predictions are triggering the exploration of novel 2D geometries [2-10], such as graphynes, Kagomé and the Lieb lattice. The latter is the 2D analogue of the 3D lattice exhibited by perovskites [2]; it is a square-depleted lattice, which is characterised by a band structure featuring Dirac cones intersected by a flat band. Whereas photonic and cold-atom Lieb lattices have been demonstrated [11-17], an electronic equivalent in 2D is difficult to realize in an existing material. Here, we report an electronic Lieb lattice formed by the surface state electrons of Cu(111) confined by an array of CO molecules positioned with a scanning tunneling microscope (STM). Using scanning tunneling microscopy, spectroscopy and wave-function mapping, we confirm the predicted characteristic electronic structure of the Lieb lattice. The experimental findings are corroborated by muffin-tin and tight-binding calculations. At higher energies, second-order electronic patterns are observed, which are equivalent to a super-Lieb lattice.

Generation of Anisotropic Massless Dirac Fermions and Asymmetric Klein Tunneling in Few-Layer Black Phosphorus Superlattices

Artificial lattices have been employed in a broad range of two-dimensional systems, including those with electrons, atoms, and photons, in the quest for massless Dirac fermions with high flexibility and controllability. Establishing triangular or hexagonal symmetry, from periodically patterned molecule assembly or electrostatic gating as well as from moiré pattern induced by substrate, has produced electronic states with linear dispersions from two-dimensional electron gas (2DEG) residing in semiconductors, metals, and graphene. Different from the commonly studied isotropic host systems, here we demonstrate that massless Dirac fermions with tunable anisotropic characteristics can, in general, be generated in highly anisotropic 2DEG under slowly varying external periodic potentials. In the case of patterned few-layer black phosphorus superlattices, the new chiral quasiparticles exist exclusively in certain isolated energy window and inherit the strong anisotropic properties of pristine black phosphorus. These states exhibit asymmetric Klein tunneling, in which the transmission probability of the wave packets with normal incidence is no longer unity and can be tuned and controlled. In general, the direction of wave packet incidence for perfect transmission and that of the normal incidence are different, and the difference can reach more than 50° under an appropriate barrier orientation in black phosphorus superlattices. Our findings provide insight into the understanding and possible utilization of these novel emergent chiral quasiparticles.

Propagation of Structural Disorder in Epitaxially Connected Quantum Dot Solids from Atomic to Micron Scale

Epitaxially connected superlattices of self-assembled colloidal quantum dots present a promising route toward exquisite control of electronic structure through precise hierarchical structuring across multiple length scales. Here, we uncover propagation of disorder as an essential feature in these systems, which intimately connects order at the atomic, superlattice, and grain scales. Accessing theoretically predicted exotic electronic states and highly tunable minibands will therefore require detailed understanding of the subtle interplay between local and long-range structure. To that end, we developed analytical methods to quantitatively characterize the propagating disorder in terms of a real paracrystal model and directly observe the dramatic impact of angstrom scale translational disorder on structural correlations at hundreds of nanometers. Using this framework, we discover improved order accompanies increasing sample thickness and identify the substantial effect of small fractions of missing epitaxial bonds on statistical disorder. These results have significant experimental and theoretical implications for the elusive goals of long-range carrier delocalization and true miniband formation.

Designing artificial two dimensional electron lattice on metal surface: a Kagome-like lattice as an example

Recently, a new kind of artificial two dimensional (2D) electron lattice on the nanoscale, i.e. molecular graphene, has drawn a lot of interest, where the metal surface electrons are transformed into a honeycomb lattice via absorbing a molecular lattice on the metal surface [Gomes et al., Nature, 2012, 438, 306; Wang et al., Phys. Rev. Lett., 2014, 113, 196803]. In this work, we theoretically demonstrate that this technique can be readily used to build other complex 2D electron lattices on a metal surface, which are of high interest in the field of condensed matter physics. The main challenge to build a complex 2D electron lattice is that this is a quantum antidot system, where the absorbed molecule normally exerts a repulsive potential on the surface electrons. Thus, there is no straightforward corresponding relation between the molecular lattice pattern and the desired 2D lattice of surface electrons. Here, we give an interesting example about the Kagome lattice, which has exotic correlated electronic states. We design a special molecular pattern and show that this molecular lattice can transform the surface electrons into a Kagome-like lattice. The numerical simulation is conducted using a Cu(111) surface and CO molecules. We first estimate the effective parameters of the Cu/CO system by fitting experimental data of the molecular graphene. Then, we calculate the corresponding energy bands and LDOS of the surface electrons in the presence of the proposed molecular lattice. Finally, we interpret the numerical results by the tight binding model of the Kagome lattice. We hope that our work can stimulate further theoretical and experimental interest in this novel artificial 2D electron lattice system.

Topological Properties of Electrons in Honeycomb Lattice with Detuned Hopping Energy

Honeycomb lattice can support electronic states exhibiting Dirac energy dispersion, with graphene as the icon. We propose to derive nontrivial topology by grouping six neighboring sites of honeycomb lattice into hexagons and enhancing the inter-hexagon hopping energies over the intra-hexagon ones. We reveal that this manipulation opens a gap in the energy dispersion and drives the system into a topological state. The nontrivial topology is characterized by the index associated with a pseudo time-reversal symmetry emerging from the C₆ symmetry of the hopping texture, where the angular momentum of orbitals accommodated on the hexagonal “artificial atoms” behaves as the pseudospin. The size of topological gap is proportional to the hopping-energy difference, which can be larger than typical spin-orbit couplings by orders of magnitude and potentially renders topological electronic transports available at high temperatures.

High-mobility capacitively-induced two-dimensional electrons in a lateral superlattice potential

In the presence of a lateral periodic potential modulation, two-dimensional electrons may exhibit interesting phenomena, such as a graphene-like energy-momentum dispersion, Bloch oscillations, or the Hofstadter butterfly band structure. To create a sufficiently strong potential modulation using conventional semiconductor heterostructures, aggressive device processing is often required, unfortunately resulting in strong disorder that masks the sought-after effects. Here, we report a novel fabrication process flow for imposing a strong lateral potential modulation onto a capacitively induced two-dimensional electron system, while preserving the host material quality. Using this process flow, the electron density in a patterned Si/SiGe heterostructure can be tuned over a wide range, from 4.4 × 10(10) cm(-2) to 1.8 × 10(11) cm(-2), with a peak mobility of 6.4 × 10(5) cm(2)/V·s. The wide density tunability and high electron mobility allow us to observe sequential emergence of commensurability oscillations as the density, the mobility, and in turn the mean free path, increase. Magnetic-field-periodic quantum oscillations associated with various closed orbits also emerge sequentially with increasing density. We show that, from the density dependence of the quantum oscillations, one can directly extract the steepness of the imposed superlattice potential. This result is then compared to a conventional lateral superlattice model potential.

Hidden symmetry and protection of Dirac points on the honeycomb lattice

The honeycomb lattice possesses a novel energy band structure, which is characterized by two distinct Dirac points in the Brillouin zone, dominating most of the physical properties of the honeycomb structure materials. However, up till now, the origin of the Dirac points is unclear yet. Here, we discover a hidden symmetry on the honeycomb lattice and prove that the existence of Dirac points is exactly protected by such hidden symmetry. Furthermore, the moving and merging of the Dirac points and a quantum phase transition, which have been theoretically predicted and experimentally observed on the honeycomb lattice, can also be perfectly explained by the parameter dependent evolution of the hidden symmetry.

Finite-size effects and interactions in artificial graphene formed by repulsive scatterers

We carry out a numerical real-space study on electrons confined in a two-dimensional triangular lattice of repulsive scattering centres. The system represents a qualitative model of molecular graphene, where the electron gas is confined between the scattering molecules in a hexagonal configuration. Our main interest is, on one hand, in the comparability of a finite system (flake) and a fully periodic one, and, on the other hand, in the role of the Coulombic electron-electron interactions and the relative strength of the scattering centres. Our real-space study shows in detail how the density of states of the fully periodic system-containing the Dirac point-is gradually formed as the size of the flake is increased. Good qualitative agreement with the experimental density of states is obtained. Our study confirms the minor role of the electron-electron interactions with selected system parameters, and shows in detail that large scattering amplitudes are required to obtain a distinctive Dirac point in the density of states.

Artificial Gauge Field and Topological Phase in a Conventional Two-dimensional Electron Gas with Antidot Lattices

Based on the Born-Oppemheimer approximation, we divide the total electron Hamiltonian in a spin-orbit coupled system into the slow orbital motion and the fast interband transition processes. We find that the fast motion induces a gauge field on the slow orbital motion, perpendicular to the electron momentum, inducing a topological phase. From this general designing principle, we present a theory for generating artificial gauge field and topological phase in a conventional two-dimensional electron gas embedded in parabolically graded GaAs/InxGa1-xAs/GaAs quantum wells with antidot lattices. By tuning the etching depth and period of the antidot lattices, the band folding caused by the antidot potential leads to the formation of minibands and band inversions between neighboring subbands. The intersubband spin-orbit interaction opens considerably large nontrivial minigaps and leads to many pairs of helical edge states in these gaps.

Two-Dimensional Superlattices of Bi Nanoclusters Formed on a Au(111) Surface Using Porous Supramolecular Templates

We used porous supramolecular structures as templates to make two-dimensional (2D) superlattices of Bi nanoclusters on a Au(111) surface. First, we applied on-surface self-assembly to prepare 2D porous supramolecular structures containing well-ordered nanopores. Then, we deposited Bi atoms on the surface. The Bi atoms were confined in the supramolecular pores and formed nanoclusters of a critical size that is defined by the pore size. These nanoclusters were arranged as a 2D superlattice dictated by the structure of the supramolecular templates. The nanocluster size and superlattice periodicity can be adjusted by appropriately designing the supramolecular structures. We further studied the formation mechanism of the nanoclusters. We found that Bi atoms could diffuse across the pore boundaries at room temperature and nucleated as clusters inside the pores. The clusters grew until they reached the critical size and became stable. We used kinetic Monte Carlo simulations to reproduce the experimental results and quantified the interpore diffusion barrier to be 0.65 eV.

Topological states in multi-orbital HgTe honeycomb lattices

Research on graphene has revealed remarkable phenomena arising in the honeycomb lattice. However, the quantum spin Hall effect predicted at the K point could not be observed in graphene and other honeycomb structures of light elements due to an insufficiently strong spin-orbit coupling. Here we show theoretically that 2D honeycomb lattices of HgTe can combine the effects of the honeycomb geometry and strong spin-orbit coupling. The conduction bands, experimentally accessible via doping, can be described by a tight-binding lattice model as in graphene, but including multi-orbital degrees of freedom and spin-orbit coupling. This results in very large topological gaps (up to 35 meV) and a flattened band detached from the others. Owing to this flat band and the sizable Coulomb interaction, honeycomb structures of HgTe constitute a promising platform for the observation of a fractional Chern insulator or a fractional quantum spin Hall phase.

Dirac point movement and topological phase transition in patterned graphene

The honeycomb lattice of graphene is characterized by linear dispersion and pseudospin chirality of fermions on the Dirac cones. If lattice anisotropy is introduced, the Dirac cones stay intact but move in reciprocal space. Dirac point movement can lead to a topological transition from semimetal to semiconductor when two inequivalent Dirac points merge, an idea that has attracted significant research interest. However, such movement normally requires unrealistically high lattice anisotropy. Here we show that anisotropic defects can break the C3 symmetry of graphene, leading to Dirac point drift in the Brillouin zone. Additionally, the long-range order in periodically patterned graphene can induce intervalley scattering between two inequivalent Dirac points, resulting in a semimetal-to-insulator topological phase transition. The magnitude and direction of Dirac point drift are predicted analytically, which are consistent with our first-principles electronic structure calculations. Thus, periodically patterned graphene can be used to study the fascinating physics associated with Dirac point movement and the corresponding phase transition.

Unveiling pseudospin and angular momentum in photonic graphene

Pseudospin, an additional degree of freedom inherent in graphene, plays a key role in understanding many fundamental phenomena such as the anomalous quantum Hall effect, electron chirality and Klein paradox. Unlike the electron spin, the pseudospin was traditionally considered as an unmeasurable quantity, immune to Stern-Gerlach-type experiments. Recently, however, it has been suggested that graphene pseudospin is a real angular momentum that might manifest itself as an observable quantity, but so far direct tests of such a momentum remained unfruitful. Here, by selective excitation of two sublattices of an artificial photonic graphene, we demonstrate pseudospin-mediated vortex generation and topological charge flipping in otherwise uniform optical beams with Bloch momentum traversing through the Dirac points. Corroborated by numerical solutions of the linear massless Dirac-Weyl equation, we show that pseudospin can turn into orbital angular momentum completely, thus upholding the belief that pseudospin is not merely for theoretical elegance but rather physically measurable.

The rare two-dimensional materials with Dirac cones

Inspired by the great development of graphene, more and more research has been conducted to seek new two-dimensional (2D) materials with Dirac cones. Although 2D Dirac materials possess many novel properties and physics, they are rare compared with the numerous 2D materials. To provide explanation for the rarity of 2D Dirac materials as well as clues in searching for new Dirac systems, here we review the recent theoretical aspects of various 2D Dirac materials, including graphene, silicene, germanene, graphynes, several boron and carbon sheets, transition-metal oxides (VO2)n/(TiO2)m and (CrO2)n/(TiO2)m, organic and organometallic crystals, so-MoS2, and artificial lattices (electron gases and ultracold atoms). Their structural and electronic properties are summarized. We also investigate how Dirac points emerge, move, and merge in these systems. The von Neumann–Wigner theorem is used to explain the scarcity of Dirac cones in 2D systems, which leads to rigorous requirements on the symmetry, parameters, Fermi level, and band overlap of materials to achieve Dirac cones. Connections between existence of Dirac cones and the structural features are also discussed.

Scalable tight-binding model for graphene

Artificial graphene consisting of honeycomb lattices other than the atomic layer of carbon has been shown to exhibit electronic properties similar to real graphene. Here, we reverse the argument to show that transport properties of real graphene can be captured by simulations using "theoretical artificial graphene." To prove this, we first derive a simple condition, along with its restrictions, to achieve band structure invariance for a scalable graphene lattice. We then present transport measurements for an ultraclean suspended single-layer graphene pn junction device, where ballistic transport features from complex Fabry-Pérot interference (at zero magnetic field) to the quantum Hall effect (at unusually low field) are observed and are well reproduced by transport simulations based on properly scaled single-particle tight-binding models. Our findings indicate that transport simulations for graphene can be efficiently performed with a strongly reduced number of atomic sites, allowing for reliable predictions for electric properties of complex graphene devices. We demonstrate the capability of the model by applying it to predict so-far unexplored gate-defined conductance quantization in single-layer graphene.

Manipulation and characterization of aperiodical graphene structures created in a two-dimensional electron gas

We demonstrate that Dirac fermions can be created and manipulated in a two-dimensional electron gas (2DEG). Using a cryogenic scanning tunneling microscope, we arranged coronene molecules one by one on a Cu(111) surface to construct artificial graphene nanoribbons with perfect zigzag (ZGNRs) or arm-chairedges and confirmed that new states localized along the edges emerge only in the ZGNRs. We further made and studied several typical defects, such as single vacancies, Stone-Wales defects, and dislocation lines, and found that all these defects introduce localized states at or near the Dirac point in the quasiparticle spectra. Our results confirm that artificial systems built on a 2DEG provide rigorous experimental verifications for several long-sought theoretical predications of aperiodic graphene structures.

From semiconductor nanocrystals to artificial solids with dimensionality below two

Two-dimensional films of semiconductors can be patterned into super-lattices with nanoscale periodicity, using top-down (lithography) or bottom-up approaches. In particular, square and honeycomb lattices of semiconductor nanocrystals have been recently synthesized using oriented attachment. We have performed atomistic tight-binding calculations of the conduction bands of super-lattices of CdSe. We consider spherical nanocrystals connected by horizontal cylinders and we investigate the band structure between two extreme limits, the uniform two-dimensional film, and the assembly of disconnected nanocrystals. Using this model system, we explain how rich band structures emerge from the periodic nano-geometry, including Dirac cones and non-trivial flat bands in honeycomb lattices. The possibility to build non-conventional band structures using multi-orbital artificial atoms (nanocrystals) opens up new prospects.

Control over band structure and tunneling in bilayer graphene induced by velocity engineering

The band structure and transport properties of massive Dirac fermions in bilayer graphene with velocity modulation in space are investigated in the presence of a previously created band gap. It is pointed out that velocity engineering may be considered as a factor to control the band gap of symmetry-broken bilayer graphene. The band gap is direct and independent of velocity value if the velocity modulated in two layers is set up equally. Otherwise, in the case of interlayer asymmetric velocity, not only is the band gap indirect, but also the electron-hole symmetry fails. This band gap is controllable by the ratio of the velocity modulated in the upper layer to the velocity modulated in the lower layer. In more detail, the shift of momentum from the conduction band edge to the valence band edge can be engineered by the gate bias and velocity ratio. A transfer matrix method is also elaborated to calculate the four-band coherent conductance through a velocity barrier possibly subjected to a gate bias. Electronic transport depends on the ratio of velocity modulated inside the barrier to that for surrounding regions. As a result, a quantum version of total internal reflection is observed for thick enough velocity barriers. Moreover, a transport gap originating from the applied gate bias is engineered by modulating the velocities of the carriers in the upper and lower layers.

Artificial honeycomb lattices for electrons, atoms and photons

Artificial honeycomb lattices offer a tunable platform for studying massless Dirac quasiparticles and their topological and correlated phases. Here we review recent progress in the design and fabrication of such synthetic structures focusing on nanopatterning of two-dimensional electron gases in semiconductors, molecule-by-molecule assembly by scanning probe methods and optical trapping of ultracold atoms in crystals of light. We also discuss photonic crystals with Dirac cone dispersion and topologically protected edge states. We emphasize how the interplay between single-particle band-structure engineering and cooperative effects leads to spectacular manifestations in tunnelling and optical spectroscopies.

Resonant tunneling in graphene pseudomagnetic quantum dots

Realistic relaxed configurations of triaxially strained graphene quantum dots are obtained from unbiased atomistic mechanical simulations. The local electronic structure and quantum transport characteristics of y-junctions based on such dots are studied, revealing that the quasi-uniform pseudomagnetic field induced by strain restricts transport to Landau level- and edge state-assisted resonant tunneling. Valley degeneracy is broken in the presence of an external field, allowing the selective filtering of the valley and chirality of the states assisting in the resonant tunneling. Asymmetric strain conditions can be explored to select the exit channel of the y-junction.

Low-dimensional semiconductor superlattices formed by geometric control over nanocrystal attachment

Oriented attachment, the process in which nanometer-sized crystals fuse by atomic bonding of specific crystal facets, is expected to be more difficult to control than nanocrystal self-assembly that is driven by entropic factors or weak van der Waals attractions. Here, we present a study of oriented attachment of PbSe nanocrystals that counteract this tuition. The reaction was studied in a thin film of the suspension casted on an immiscible liquid at a given temperature. We report that attachment can be controlled such that it occurs with one type of facets exclusively. By control of the temperature and particle concentration we obtain one- or two-dimensional PbSe single crystals, the latter with a honeycomb or square superimposed periodicity in the nanometer range. We demonstrate the ability to convert these PbSe superstructures into other semiconductor compounds with the preservation of crystallinity and geometry.

Topological insulating states in laterally patterned ordinary semiconductors

We propose that ordinary semiconductors with large spin-orbit coupling, such as GaAs, can host stable, robust, and tunable topological states in the presence of quantum confinement and superimposed potentials with hexagonal symmetry. We show that the electronic gaps which support chiral spin edge states can be as large as the electronic bandwidth in the heterostructure miniband. The existing lithographic technology can produce a topological insulator operating at a temperature of 10-100 K. Improvement of lithographic techniques will open the way to a tunable room temperature topological insulator.

Fractional topological phases and broken time-reversal symmetry in strained graphene

We show that strained or deformed honeycomb lattices are promising platforms to realize fractional topological quantum states in the absence of any magnetic field. The strain-induced pseudomagnetic fields are oppositely oriented in the two valleys and can be as large as 60-300 T as reported in recent experiments. For strained graphene at neutrality, a spin- or a valley-polarized state is predicted depending on the value of the on-site Coulomb interaction. At fractional filling, the unscreened Coulomb interaction leads to a valley-polarized fractional quantum Hall liquid which spontaneously breaks time-reversal symmetry. Motivated by artificial graphene systems, we consider tuning the short-range part of interactions and demonstrate that exotic valley symmetric states, including a valley fractional topological insulator and a spin triplet superconductor, can be stabilized by such interaction engineering.

Electron-electron interactions in artificial graphene

Recent advances in the creation and modulation of graphenelike systems are introducing a science of "designer Dirac materials". In its original definition, artificial graphene is a man-made nanostructure that consists of identical potential wells (quantum dots) arranged in an adjustable honeycomb lattice in the two-dimensional electron gas. As our ability to control the quality of artificial graphene samples improves, so grows the need for an accurate theory of its electronic properties, including the effects of electron-electron interactions. Here we determine those effects on the band structure and on the emergence of Dirac points.

Space dependent Fermi velocity in strained graphene

We investigate some apparent discrepancies between two different models for curved graphene: the one based on tight-binding and elasticity theory, and the covariant approach based on quantum field theory in curved space. We demonstrate that strained or corrugated samples will have a space-dependent Fermi velocity in either approach that can affect the interpretation of local probe experiments in graphene. We also generalize the tight-binding approach to general inhomogeneous strain and find a gauge field proportional to the derivative of the strain tensor that has the same form as the one obtained in the covariant approach.

Designer Dirac fermions and topological phases in molecular graphene

The observation of massless Dirac fermions in monolayer graphene has generated a new area of science and technology seeking to harness charge carriers that behave relativistically within solid-state materials. Both massless and massive Dirac fermions have been studied and proposed in a growing class of Dirac materials that includes bilayer graphene, surface states of topological insulators and iron-based high-temperature superconductors. Because the accessibility of this physics is predicated on the synthesis of new materials, the quest for Dirac quasi-particles has expanded to artificial systems such as lattices comprising ultracold atoms. Here we report the emergence of Dirac fermions in a fully tunable condensed-matter system-molecular graphene-assembled by atomic manipulation of carbon monoxide molecules over a conventional two-dimensional electron system at a copper surface. Using low-temperature scanning tunnelling microscopy and spectroscopy, we embed the symmetries underlying the two-dimensional Dirac equation into electron lattices, and then visualize and shape the resulting ground states. These experiments show the existence within the system of linearly dispersing, massless quasi-particles accompanied by a density of states characteristic of graphene. We then tune the quantum tunnelling between lattice sites locally to adjust the phase accrual of propagating electrons. Spatial texturing of lattice distortions produces atomically sharp p-n and p-n-p junction devices with two-dimensional control of Dirac fermion density and the power to endow Dirac particles with mass. Moreover, we apply scalar and vector potentials locally and globally to engender topologically distinct ground states and, ultimately, embedded gauge fields, wherein Dirac electrons react to 'pseudo' electric and magnetic fields present in their reference frame but absent from the laboratory frame. We demonstrate that Landau levels created by these gauge fields can be taken to the relativistic magnetic quantum limit, which has so far been inaccessible in natural graphene. Molecular graphene provides a versatile means of synthesizing exotic topological electronic phases in condensed matter using tailored nanostructures.