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Layered van der Waals transition metal dichalcogenides (TMDCs), generally composed of three atomic X-M-X planes in each layer (M = transition metal, X = chalcogen), provide versatile platforms for exploring diverse quantum phenomena. In each MX2 layer, the M-X bonds are predominantly covalent in nature, as a result, the cleavage of TMDC crystals always occurring between the layers. Here we report the controllable realization of fractional-layer WTe2 via an in-situ scanning tunnelling microscopy (STM) tip manipulation technique. By applying STM tip pulses, hundreds of the topmost Te atoms are removed to form a nanoscale monolayer Te pit in the 1T'-WTe2, thus realizing a brand-new 2/3-layer WTe2. Such a unique configuration undergoes a spontaneous atomic reconstruction, yielding an energy-dependent unidirectional charge-density-wave state with the wavevector and geometry quite distinct from that of pristine 1T'-WTe2. Our results expand the conventional understanding of the TMDCs and are expected to stimulate the research on extraordinary structures and properties based on fractional-layer TMDCs., Comment: 4 figures in main text

Phase singularities are phase-indeterminate points where wave amplitudes are zero, which manifest as phase vertices or wavefront dislocations. In the realm of optical and electron beams, the phase singularity has been extensively explored, demonstrating a profound connection to orbital angular momentum. Direct local imaging of the impact of orbital angular momentum on phase singularities at the nanoscale, however, remains a challenge and has yet to be achieved. Here, we study the role of orbital angular momentum in phase singularities in graphene, particularly at the atomic level, through scanning tunneling microscopy and spectroscopy. Our experiments demonstrate that the scatterings between different orbital angular momentum states, which are induced by local rotational symmetry-breaking potentials, can generate additional phase singularity, and result in robust single wavefront dislocation in real space. Our results pave the way for exploring the effects of orbital degree of freedom on quantum phases in quasiparticle interference processes., Comment: 28 pages, 3 figures, 10 extended figures

In a molecule formed by two atoms, energy difference between bonding and antibonding orbitals should depend on distance of the two atoms. However, exploring molecular orbitals of two natural atoms with tunable distance has remained an outstanding experimental challenge. Graphene quantum dots (GQDs) can be viewed as relativistic artificial atoms, therefore, offering a unique platform to study molecular physics. Here, through scanning tunneling microscope (STM), we create and directly visualize the formation process of relativistic artificial molecules based on two coupled GQDs with tunable distance. Our study indicates that energy difference between the bonding and antibonding orbitals of the lowest quasibound state increases linearly with inverse distance of the two GQDs due to the relativistic nature of the artificial molecule. For quasibound states with higher orbital momenta, the coupling between these states leads to half-energy spacing of the confined states because the length of the molecular-like orbit is about twice that of the atomic-like orbit. Evolution from ring-like whispering-gallery modes in the artificial atoms to figure-eight orbitals in the artificial molecules is directly imaged. The ability to resolve the coupling and orbitals of the relativistic artificial molecule at the nanoscale level yields insights into the behavior of quantum-relativistic matter.

It has long been an ultimate goal to introduce chemical doping at the atomic level to precisely tune properties of materials. Two-dimensional materials have natural advantage because of its highly-exposed surface atoms, however, it is still a grand challenge to achieve this goal experimentally. Here, we demonstrate the ability to introduce chemical doping in graphene with atomic-level precision by controlling chemical adsorption of individual Se atoms, which are extracted from the underneath WSe2, at the interface of graphene-WSe2 heterostructures. Our scanning tunneling microscopy (STM) measurements, combined with first-principles calculations, reveal that individual Se atoms can chemisorbed on three possible positions in graphene, which generate distinct pseudospin-mediated atomic-scale vortices in graphene. We demonstrate that the chemisorbed positions of individual Se atoms can be manipulated by STM tip, which enables us to achieve atomic-scale controlling quantum interference of the pseudospin-mediated vortices in graphene. This result offers the promise of controlling properties of materials through chemical doping with atomic-level precision.

In twisted bilayer graphene (TBG), a twist-angle-dependent competition between interlayer stacking energy and intralayer elastic energy results in flat rigid layers at large twist angles and lattice reconstruction at small twist angles. Despite enormous scientific interest and effort in the TBG, however, an experimental study of evolution from the rigid lattice to the reconstructed lattice as a function of twist angle is still missing. Here we present a scanning tunneling microscopy and spectroscopy study to reveal the twist-angle-dependent lattice reconstruction in the TBG. Our experiment demonstrates that there is a transition regime between the rigid regime and the relaxed regime and, unexpectedly, the reconstructed and unreconstructed structures coexist in the transition regime. The coexistence of the two distinct structures in this regime may arise from subtle balance between the interlayer stacking energy and intralayer elastic energy in the TBG with intermediate moir\'e sizes., Comment: 4 Figures in main text

Kekul\'e phases are Peierls-like lattice distortions in graphene that are predicted to host novel electronic states beyond graphene (1-8). Although the Kekul\'e phases are realized in graphene through introducing electron-electron interactions at high magnetic fields (9-11) or adatom superlattices (12-15), it is still an extremely challenge to obtain large-area graphene Kekul\'e phases in experiment. Here we demonstrate that sample-wide Kekul\'e distortions in graphene can be stabilized by using transition metal dichalcogenides (TMDs) as substrates and the induced Kekul\'e orders are quite robust in the whole graphene/TMDs heterostructures with different twist angles. The commensurate structures of the heterostructures provide periodic scattering centers that break the translational symmetry of graphene and couple electrons of the two valleys in graphene, which tips the graphene toward global Kekul\'e density wave phases. Unexpectedly, three distinct Kekul\'e bond textures stabilized at various energies are directly imaged in every graphene/TMDs heterostructure. Our results reveal an unexpected sensitivity of electronic properties in graphene to the supporting substrates and provide an attractive route toward designing novel phases in graphene/TMDs heterostructures.

Overlaying two graphene layers with a small twist angle can create a moire superlattice to realize exotic phenomena that are entirely absent in graphene monolayer. A representative example is the predicted formation of localized pseudo-Landau levels (PLLs) with Kagome lattice in tiny-angle twisted bilayer graphene (TBG) with theta < 0.3 deg when the graphene layers are subjected to different electrostatic potentials. However, this was shown only for the model of rigidly rotated TBG which is not realized in reality due to an interfacial structural reconstruction. It is believed that the interfacial structural reconstruction strongly inhibits the formation of the PLLs. Here, we systematically study electronic properties of the TBG with 0.075 deg < theta < 1.2 deg and demonstrate, unexpectedly, that the PLLs are quite robust for all the studied TBG. The structural reconstruction suppresses the formation of the emergent Kagome lattice in the tiny-angle TBG. However, for the TBG around magic angle, the sample-wide electronic Kagome lattices with tunable lattice constants are directly imaged by using scanning tunneling microscope. Our observations open a new direction to explore exotic correlated phases in moire systems., Comment: 4 figures in main text. PRL in press

Electronic properties of quantum dots (QDs) depend sensitively on their parent materials. Therefore, confined electronic states in graphene QDs (GQDs) of monolayer and Bernal-stacked bilayer graphene are quite different. Twisted bilayer graphene (TBG) is distinct from monolayer and Bernal-stacked bilayer graphene because of the new degree of freedom: twist angle. In the past few years, numerous efforts have been made to realize the GQDs of monolayer and Bernal-stacked bilayer graphene and achieved great success. Thus far, however, strategies for realizing GQDs of TBG have been elusive. Here, we demonstrate a general approach for fabricating stationary GQDs of TBG by introducing nanoscale p-n junctions with sharp boundaries in the TBG. We verify the confinement of low-energy massless Dirac fermions via whispering-gallery modes in the GQDs of TBG. Unexpectedly, electronic states around van Hove singularities of the TBG are also strongly modified around the GQDs. Such a feature has never been reported and is attributed to spatial variation of the interlayer coupling in the TBG induced by the GQDs.

Physical properties of two-dimensional van der Waals (vdWs) structures depend sensitively on both stacking orders and interlayer interactions. Yet, in most cases studied to date, the interlayer interaction is considered to be a static property of the vdWs structures. Here we demonstrate that applying a scanning tunneling microscopy (STM) tip pulse on twisted bilayer graphene (TBG) can induce sub-Angstrom fluctuations of the interlayer separation in the TBG, which are equivalent to dynamic vertical external pressure of about 10 GPa on the TBG. The sub-Angstrom fluctuations of the interlayer separation result in large oscillations of the energy separations between two van Hove singularities (VHSs) in the TBG. The period of the oscillations of the VHSs spacing is extremely long, about 500-1000 seconds, attributing to tip-induced local stress in the atomic-thick TBG. Our result provides an efficient method to tune and measure the physical properties of the vdWs structures dynamically., Comment: 4 Figures in main text

The interplay between interlayer van der Waals interaction and intralayer lattice distortion can lead to structural reconstruction in slightly twisted bilayer graphene (TBG) with the twist angle being smaller than a characteristic angle {\theta}c. Experimentally, the {\theta}c is demonstrated to be very close to the magic angle ({\theta} ~ 1.05{\deg}). In this work, we address the transition between reconstructed and unreconstructed structures of the TBG across the magic angle by using scanning tunnelling microscopy (STM). Our experiment demonstrates that both the two structures are stable in the TBG around the magic angle. By applying a STM tip pulse, we show that the two structures can be switched to each other and the bandwidth of the flat bands, which plays a vital role in the emergent strongly correlated states in the magic-angle TBG, can be tuned. The observed tunable lattice reconstruction and bandwidth of the flat bands provide an extra control knob to manipulate the exotic electronic states of the TBG near the magic angle., Comment: 4 figures in main text

Let $f(n,k)$ be the largest number of positive integers not exceeding $n$ from which one cannot select $k+1$ pairwise coprime integers, and let $E(n,k)$ be the set of positive integers which do not exceed $n$ and can be divided by at least one of $p_1, p_2,..., p_k$, where $p_i$ is the $i$-th prime. In 1962, P. Erd\H os conjectured that $f(n,k)=|E(n,k)|$ for all $n\ge p_k$. In 1973, S. L. G. Choi proved that the conjecture is true for $k=3$. In 1994, Ahlswede and Kachatrian disproved the conjecture for $k=212$. In this paper we prove that, for $n\ge 49$, if A(n,4) is a set of positive integers not exceeding $n$ from which one cannot select 5 pairwise coprime integers and $|A(n,4)|\ge |E(n,4)|$, then $A(n,4)=E(n,4)$. In particular, the conjecture is true for k=4. Several open problems and conjectures are posed for further research., Comment: 17 pages