Long-wavelength local density of states oscillations near graphene step edges

LDOS of electron pair and the role of the Pauli exclusion principle

The local density of states (LDOS) for a pair of non-relativistic electrons, influenced by repulsive Coulomb forces, is expressed in term of one-dimensional integrals over Whittaker functions. The computation of the electron pair's LDOS relies on a two-particle Green's function (GF), a generalization of the one-particle GF applicable to a charged particle in an attractive Coulomb potential. By incorporating electron spins and considering the Pauli exclusion principle, the resulting LDOS consists of two components: one originating from an exchange-even two-particle GF and the other from an exchange-odd two-particle GF. The calculated LDOS reveals its dependence on both inter-electron distance and energy. The pseudo-LDOS, derived from the two-body contribution to the LDOS, is examined. This term ensures complete LDOS suppression atr = 0, exhibiting a limited spatial extent, and the reasons for its emergence are elucidated. It is shown that for energies exceeding the effective Hartree energy and inter-electron distances beyond the effective Bohr radius, the impact of many-body contributions to the LDOS can be disregarded. The induced LDOS for an electron pair subjected to an attractive contact potential in two dimensions is evaluated. At small distancesafrom the potential center, a predicted relative difference in LDOS between even and odd state pair reaches approximately 8%. The calculated LDOS is compared with available experimental findings from a two-dimensional electron gas (2DEG). Both exhibit similar oscillation periods; however, the LDOS of the electron pair decays as1/a3, significantly faster than the1/adecay observed for free electrons in a 2DEG.

Analogous electronic states in graphene and planer metallic quantum dots

Graphene nanostructures offer wide range of applications due to their distinguished and tunable electronic properties. Recently, atomic and molecular graphene were modeled following simple free-electron scattering by periodic muffin tin potential leading to remarkable agreement with density functional theory. Here we extend the analogy of the π -electronic structures and quantum effects between atomic graphene quantum dots (QDs) and homogeneous planer metallic counterparts of similar size and shape. Specifically, we show that at high binding energies, below the M ¯ -point gap, graphene QDs enclose confined states and standing wave quasiparticle interference patterns analogous to those reported on coinage metal surfaces for nanoscale confining structures such as vacancy islands and quantum corrals. These confined and quantum corral-like states in graphene QDs can be resolved in tomography experiments using angle-resolved photoemission spectroscopy. Likewise, the shape of near-Fermi frontier orbitals in graphene quantum dots can be reproduced from electron confinement within homogeneous metal QDs of identical size and shape. Furthermore, confined states analogous to those found in metallic quantum stadiums can be realized in coupled QDs of graphene for reduced separation. The present study offer a simple fundamental understanding of graphene electronic structures and also open the way towards efficient modeling of novel graphene-based nanostructures.

Machine Learning for Optical Scanning Probe Nanoscopy

The ability to perform nanometer-scale optical imaging and spectroscopy is key to deciphering the low-energy effects in quantum materials, as well as vibrational fingerprints in planetary and extraterrestrial particles, catalytic substances, and aqueous biological samples. These tasks can be accomplished by the scattering-type scanning near-field optical microscopy (s-SNOM) technique that has recently spread to many research fields and enabled notable discoveries. Herein, it is shown that the s-SNOM, together with scanning probe research in general, can benefit in many ways from artificial-intelligence (AI) and machine-learning (ML) algorithms. Augmented with AI- and ML-enhanced data acquisition and analysis, scanning probe optical nanoscopy is poised to become more efficient, accurate, and intelligent.

Suppression of Intervalley Coupling in Graphene via Potassium Doping

The quantum interference patterns induced by impurities in graphene can form the (√3 × √3)R30° superlattice with intervalley scattering. This superlattice can lead to the folded Dirac cone at the center of Brillouin zone by coupling two non-equivalent valleys. Using angle-resolved photoemission spectroscopy (ARPES), we report the observation of suppression of the folded Dirac cone in mono- and bilayer graphene upon potassium doping. The intervalley coupling with chiral symmetry broken can persist upon a light potassium-doped level but be ruined at the heavily doped level. Meanwhile, the folded Dirac cone can be suppressed by the renormalization of the Dirac band with potassium doping. Our results demonstrate that the suppression of the intervalley scattering pattern by potassium doping could pave the way toward the realization of novel chiraltronic devices in superlattice graphene.

Deep Learning Analysis of Polaritonic Wave Images

Deep learning (DL) is an emerging analysis tool across the sciences and engineering. Encouraged by the successes of DL in revealing quantitative trends in massive imaging data, we applied this approach to nanoscale deeply subdiffractional images of propagating polaritonic waves in complex materials. Utilizing the convolutional neural network (CNN), we developed a practical protocol for the rapid regression of images that quantifies the wavelength and the quality factor of polaritonic waves. Using simulated near-field images as training data, the CNN can be made to simultaneously extract polaritonic characteristics and material parameters in a time scale that is at least 3 orders of magnitude faster than common fitting/processing procedures. The CNN-based analysis was validated by examining the experimental near-field images of charge-transfer plasmon polaritons at graphene/α-RuCl₃ interfaces. Our work provides a general framework for extracting quantitative information from images generated with a variety of scanning probe methods.

Scanning tunneling spectroscopy studies of topological materials

Topological materials have become promising materials for next-generation devices by utilizing their exotic electronic states. Their exotic states caused by spin-orbital coupling usually locate on the surfaces or at the edges. Scanning tunneling spectroscopy (STS) is a powerful tool to reveal the local electronic structures of condensed matters. Therefore, STS provides us with an almost perfect method to access the exotic states of topological materials. In this topical review, we report the current investigations by several methods based on the STS technique for layered topological material from transition metal dichalcogenide Weyl semimetals (WTe₂ and MoTe₂) to two dimensional topological insulators (layered bismuth and silicene). The electronic characteristics of these layered topological materials are experimentally identified.

Emergence of Fourier's Law of Heat Transport in Quantum Electron Systems

The microscopic origins of Fourier's venerable law of thermal transport in quantum electron systems has remained somewhat of a mystery, given that previous derivations were forced to invoke intrinsic scattering rates far exceeding those occurring in real systems. We propose an alternative hypothesis, namely, that Fourier's law emerges naturally if many quantum states participate in the transport of heat across the system. We test this hypothesis systematically in a graphene flake junction and show that the temperature distribution becomes nearly classical when the broadening of the individual quantum states of the flake exceeds their energetic separation. We develop a thermal resistor network model to investigate the scaling of the sample and contact thermal resistances and show that the latter is consistent with classical thermal transport theory in the limit of large level broadening.

Quasiparticle interference in unconventional 2D systems

At present, research of 2D systems mainly focuses on two kinds of materials: graphene-like materials and transition-metal dichalcogenides (TMDs). Both of them host unconventional 2D electronic properties: pseudospin and the associated chirality of electrons in graphene-like materials, and spin-valley-coupled electronic structures in the TMDs. These exotic electronic properties have attracted tremendous interest for possible applications in nanodevices in the future. Investigation on the quasiparticle interference (QPI) in 2D systems is an effective way to uncover these properties. In this review, we will begin with a brief introduction to 2D systems, including their atomic structures and electronic bands. Then, we will discuss the formation of Friedel oscillation due to QPI in constant energy contours of electron bands, and show the basic concept of Fourier-transform scanning tunneling microscopy/spectroscopy (FT-STM/STS), which can resolve Friedel oscillation patterns in real space and consequently obtain the QPI patterns in reciprocal space. In the next two parts, we will summarize some pivotal results in the investigation of QPI in graphene and silicene, in which systems the low-energy quasiparticles are described by the massless Dirac equation. The FT-STM experiments show there are two different interference channels (intervalley and intravalley scattering) and backscattering suppression, which associate with the Dirac cones and the chirality of quasiparticles. The monolayer and bilayer graphene on different substrates (SiC and metal surfaces), and the monolayer and multilayer silicene on a Ag(1 1 1) surface will be addressed. The fifth part will introduce the FT-STM research on QPI in TMDs (monolayer and bilayer of WSe₂), which allow us to infer the spin texture of both conduction and valence bands, and present spin-valley coupling by tracking allowed and forbidden scattering channels.

Fingerprints of Multiple Electron Scatterings in Single-Layer Graphene

The electrons in graphene exhibit unusual behaviours, which can be described by massless Dirac quasiparticles. Understanding electron scattering in graphene has been of significant importance for its future application in electronic devices because electron scattering determines electrical properties such as resistivity and electron transport. There are two types of electron scatterings in graphene: intervalley scattering and intravalley scattering. In single-layer graphene, to date, it has been difficult to observe intravalley scattering because of the suppression of backscattering resulting from the chiral nature of the electrons in graphene. Here, we report the multiple electron scattering behaviours in single-layer graphene on a metallic substrate. By applying one- and two-dimensional Fourier transforms to maps of the local density of states, we can distinguish individual scattering processes from complex interference patterns. These techniques enable us to provide direct evidence of intravalley scattering, revealing a linear dispersion relation with a Fermi velocity of ~7.4 × 10(5) m/s.

One-dimensional edge states with giant spin splitting in a bismuth thin film

To realize a one-dimensional (1D) system with strong spin-orbit coupling is a big challenge in modern physics, since the electrons in such a system are predicted to exhibit exotic properties unexpected from the 2D or 3D counterparts, while it was difficult to realize genuine physical properties inherent to the 1D system. We demonstrate the first experimental result that directly determines the purely 1D band structure by performing spin-resolved angle-resolved photoemission spectroscopy of Bi islands on a silicon surface that contains a metallic 1D edge structure with unexpectedly large Rashba-type spin-orbit coupling suggestive of the nontopological nature. We have also found a sizable out-of-plane spin polarization of the 1D edge state, consistent with our first-principles band calculations. Our result provides a new platform to realize exotic quantum phenomena at the 1D edge of the strong spin-orbit-coupling systems.

Graphene on hexagonal boron nitride

The field of graphene research has developed rapidly since its first isolation by mechanical exfoliation in 2004. Due to the relativistic Dirac nature of its charge carriers, graphene is both a promising material for next-generation electronic devices and a convenient low-energy testbed for intrinsically high-energy physical phenomena. Both of these research branches require the facile fabrication of clean graphene devices so as not to obscure its intrinsic physical properties. Hexagonal boron nitride has emerged as a promising substrate for graphene devices as it is insulating, atomically flat and provides a clean charge environment for the graphene. Additionally, the interaction between graphene and boron nitride provides a path for the study of new physical phenomena not present in bare graphene devices. This review focuses on recent advancements in the study of graphene on hexagonal boron nitride devices from the perspective of scanning tunneling microscopy with highlights of some important results from electrical transport measurements.

Anisotropic Fabry-Pérot resonant states confined within nano-steps on the topological insulator surface

The peculiar nature of topological surface states, such as absence of backscattering, weak anti-localization, and quantum anomalous Hall effect, has been demonstrated mainly in bulk and film of topological insulator (TI), using surface sensitive probes and bulk transport probes. However, it is equally important and experimentally challenging to confine massless Dirac fermions with nano-steps on TI surfaces. This potential structure has similar ground with linearly-dispersed photons in Fabry-Pérot resonators, while reserving fundamental differences from well-studied Fabry-Pérot resonators and quantum corrals on noble metal surfaces. In this paper, we study the massless Dirac fermions confined within steps along the x (Γ–K) or y (Γ–M) direction on the TI surface, and the Fabry-Pérot-like resonances in the electronic local density of states (LDOS) between the steps are found. Due to the remarkable warping effect in the topological surface states, the LDOS confined in the step-well running along Γ-M direction exhibit anisotropic resonance patterns as compared to those in the step-well along Γ-K direction. The transmittance properties and spin orientation of Dirac fermion in both cases are also anisotropic in the presence of warping effect.

Theoretical analysis of a dual-probe scanning tunneling microscope setup on graphene

Experimental advances allow for the inclusion of multiple probes to measure the transport properties of a sample surface. We develop a theory of dual-probe scanning tunneling microscopy using a Green's function formalism, and apply it to graphene. Sampling the local conduction properties at finite length scales yields real space conductance maps which show anisotropy for pristine graphene systems and quantum interference effects in the presence of isolated impurities. Spectral signatures in the Fourier transforms of real space conductance maps include characteristics that can be related to different scattering processes. We compute the conductance maps of graphene systems with different edge geometries or height fluctuations to determine the effects of nonideal graphene samples on dual-probe measurements.

Observation of Dirac cone warping and chirality effects in silicene

We performed low temperature scanning tunneling microscopy (STM) and spectroscopy (STS) studies on the electronic properties of (√3 × √3)R30° phase of silicene on Ag(111) surface. We found the existence of Dirac Fermion chirality through the observation of -1.5 and -1.0 power law decay of quasiparticle interference (QPI) patterns. Moreover, in contrast to the trigonal warping of Dirac cone in graphene, we found that the Dirac cone of silicene is hexagonally warped, which is further confirmed by density functional calculations and explained by the unique superstructure of silicene. Our results demonstrate that the (√3 × √3)R30° phase is an ideal system to investigate the unique Dirac Fermion properties of silicene.

Physicochemical insight into gap openings in graphene

Based on a newly developed size-dependent cohesive energy formula for two-dimensional materials, a unified theoretical model was established to illustrate the gap openings in disordered graphene flakes, involving quantum dots, nanoribbons and nanoporous sheets. It tells us that the openings are essentially dominated by the variation in cohesive energy of C atoms, associated to the edge physicochemical nature regarding the coordination imperfection or the chemical bonding. In contrast to those ideal flakes, consequently, the gaps can be opened monotonously for disordered flakes on changing their sizes, affected by the dimension, geometric shape and the edge saturation. Using the density functional theory, accordingly, the electronic structures of disordered flakes differ to the ideal case because of the edge disorder. Our theoretical predictions have been validated by available experimental results, and provide us a distinct way for the quantitative modulation of bandgap in graphene for nanoelectronics.

Atomic-scale mapping of thermoelectric power on graphene: role of defects and boundaries

The spatially resolved thermoelectric power is studied on epitaxial graphene on SiC with direct correspondence to graphene atomic structures by a scanning tunneling microscopy (STM) method. A thermovoltage arises from a temperature gradient between the STM tip and the sample, and variations of thermovoltage are distinguished at defects and boundaries with atomic resolution. The epitaxial graphene has a high thermoelectric power of 42 μV/K with a big change (9.6 μV/K) at the monolayer-bilayer boundary. Long-wavelength oscillations are revealed in thermopower maps which correspond to the Friedel oscillations of electronic density of states associated with the intravalley scattering in graphene. On the same terrace of a graphene layer, thermopower distributions show domain structures that can be attributed to the modifications of local electronic structures induced by microscopic distortions (wrinkles) of graphene sheet on the SiC substrate. The thermoelectric power, the electronic structure, the carrier concentration, and their interplay are analyzed on the level of individual defects and boundaries in graphene.

Evidence for Dirac fermions in a honeycomb lattice based on silicon

Silicene, a sheet of silicon atoms in a honeycomb lattice, was proposed to be a new Dirac-type electron system similar to graphene. We performed scanning tunneling microscopy and spectroscopy studies on the atomic and electronic properties of silicene on Ag(111). An unexpected √3 × √3 reconstruction was found, which is explained by an extra-buckling model. Pronounced quasiparticle interferences (QPI) patterns, originating from both the intervalley and intravalley scatter, were observed. From the QPI patterns we derived a linear energy-momentum dispersion and a large Fermi velocity, which prove the existence of Dirac fermions in silicene.