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The quantum Hall (QH) effect, a topologically non-trivial quantum phase, expanded and brought into focus the concept of topological order in physics. The topologically protected quantum Hall edge states are of crucial importance to the QH effect but have been measured with limited success. The QH edge states in graphene take on an even richer role as graphene is distinguished by its four-fold degenerate zero energy Landau level (zLL), where the symmetry is broken by electron interactions on top of lattice-scale potentials but has eluded spatial measurements. In this report, we map the quantum Hall broken-symmetry edge states comprising the graphene zLL at integer filling factors of $\nu=0,\pm 1$ across the quantum Hall edge boundary using atomic force microscopy (AFM). Measurements of the chemical potential resolve the energies of the four-fold degenerate zLL as a function of magnetic field and show the interplay of the moir\'e superlattice potential of the graphene/boron nitride system and spin/valley symmetry-breaking effects in large magnetic fields.

Neuromorphic networks based on nanodevices, such as metal oxide memristors, phase change memories, and flash memory cells, have generated considerable interest for their increased energy efficiency and density in comparison to graphics processing units (GPUs) and central processing units (CPUs). Though immense acceleration of the training process can be achieved by leveraging the fact that the time complexity of training does not scale with the network size, it is limited by the space complexity of stochastic gradient descent, which grows quadratically. The main objective of this work is to reduce this space complexity by using low-rank approximations of stochastic gradient descent. This low spatial complexity combined with streaming methods allows for significant reductions in memory and compute overhead, opening the doors for improvements in area, time and energy efficiency of training. We refer to this algorithm and architecture to implement it as the streaming batch eigenupdate (SBE) approach., Comment: 13 pages, 5 figures

Quantum-relativistic matter is ubiquitous in nature; however it is notoriously difficult to probe. The ease with which external electric and magnetic fields can be introduced in graphene opens a door to creating a table-top prototype of strongly confined relativistic matter. Here, through a detailed spectroscopic mapping, we provide a spatial visualization of the interplay between spatial and magnetic confinement in a circular graphene resonator. We directly observe the development of a multi-tiered "wedding cake"-like structure of concentric regions of compressible/incompressible quantum Hall states, a signature of electron interactions in the system. Solid-state experiments can therefore yield insights into the behaviour of quantum-relativistic matter under extreme conditions.

We report a rectangular charge density wave (CDW) phase in strained 1T-VSe$_2$ thin films synthesized by molecular beam epitaxy on c-sapphire substrates. The observed CDW structure exhibits an unconventional rectangular 4a{\times}{\sqrt{3a}} periodicity, as opposed to the previously reported hexagonal $4a\times4a$ structure in bulk crystals and exfoliated thin layered samples. Tunneling spectroscopy shows a strong modulation of the local density of states of the same $4a\times\sqrt{3}a$ CDW periodicity and an energy gap of $2\Delta_{CDW}=(9.1\pm0.1)$ meV. The CDW energy gap evolves into a full gap at temperatures below 500 mK, indicating a transition to an insulating phase at ultra-low temperatures. First-principles calculations confirm the stability of both $4a\times4a$ and $4a\times\sqrt{3}a$ structures arising from soft modes in the phonon dispersion. The unconventional structure becomes preferred in the presence of strain, in agreement with experimental findings.

The phase of a quantum state may not return to its original value after the system's parameters cycle around a closed path; instead, the wavefunction may acquire a measurable phase difference called the Berry phase. Berry phases typically have been accessed through interference experiments. Here, we demonstrate an unusual Berry-phase-induced spectroscopic feature: a sudden and large increase in the energy of angular-momentum states in circular graphene p-n junction resonators when a small critical magnetic field is reached. This behavior results from turning on a $\pi$-Berry phase associated with the topological properties of Dirac fermions in graphene. The Berry phase can be switched on and off with small magnetic field changes on the order of 10 mT, potentially enabling a variety of optoelectronic graphene device applications.

Metal oxide resistive switches are increasingly important as possible artificial synapses in next generation neuromorphic networks. Nevertheless, there is still no codified set of tools for studying properties of the devices. To this end, we demonstrate electron beam induced current measurements as a powerful method to monitor the development of local resistive switching in TiO2 based devices. By comparing beam-energy dependent electron beam induced currents with Monte Carlo simulations of the energy absorption in different device layers, it is possible to deconstruct the origins of filament image formation and relate this to both morphological changes and the state of the switch. By clarifying the contrast mechanisms in electron beam induced current microscopy it is possible to gain new insights into the scaling of the resistive switching phenomenon and observe the formation of a current leakage region around the switching filament. Additionally, analysis of symmetric device structures reveals propagating polarization domains., Comment: 27 Pages 10 figures

Electron beam induced current (EBIC) is a powerful characterization technique which offers the high spatial resolution needed to study polycrystalline solar cells. Current models of EBIC assume that excitations in the $p$-$n$ junction depletion region result in perfect charge collection efficiency. However we find that in CdTe and Si samples prepared by focused ion beam (FIB) milling, there is a reduced and nonuniform EBIC lineshape for excitations in the depletion region. Motivated by this, we present a model of the EBIC response for excitations in the depletion region which includes the effects of surface recombination from both charge-neutral and charged surfaces. For neutral surfaces we present a simple analytical formula which describes the numerical data well, while the charged surface response depends qualitatively on the location of the surface Fermi level relative to the bulk Fermi level. We find the experimental data on FIB-prepared Si solar cells is most consistent with a charged surface, and discuss the implications for EBIC experiments on polycrystalline materials., Comment: 7 pages, 8 figures

We report on spatial measurements of the superconducting proximity effect in epitaxial graphene induced by a graphene-superconductor interface. Superconducting aluminum films were grown on epitaxial multilayer graphene on SiC. The aluminum films were discontinuous with networks of trenches in the film morphology reaching down to exposed graphene terraces. Scanning tunneling spectra measured on the graphene terraces show a clear decay of the superconducting energy gap with increasing separation from the graphene-aluminum edges. The spectra were well described by Bardeen-Cooper-Schrieffer (BCS) theory. The decay length for the superconducting energy gap in graphene was determined to be greater than 400 nm. Deviations in the exponentially decaying energy gap were also observed on a much smaller length scale of tens of nanometers.

Superconductivity results from a Bose condensate of Cooper-paired electrons with a macroscopic quantum wavefunction. Dramatic effects can occur when the region of the condensate is shaped and confined to the nanometer scale. Recent progress in nanostructured superconductors has revealed a route to topological superconductivity, with possible applications in quantum computing. However, challenges remain in controlling the shape and size of specific superconducting materials. Here, we report a new method to create nanostructured superconductors by partial crystallization of the half-Heusler material, YPtBi. Superconducting islands, with diameters in the range of 100 nm, were reproducibly created by local current annealing of disordered YPtBi in the tunneling junction of a scanning tunneling microscope (STM). We characterize the superconducting island properties by scanning tunneling spectroscopic measurements to determine the gap energy, critical temperature and field, coherence length, and vortex formations. These results show unique properties of a confined superconductor and demonstrate that this new method holds promise to create tailored superconductors for a wide variety of nanometer scale applications., Comment: 12 figures

Recent experiments reveal that a scanning tunneling microscopy (STM) probe tip can generate a highly localized strain field in a graphene drumhead, which in turn leads to pseudomagnetic fields in the graphene that can spatially confine graphene charge carriers in a way similar to a lithographically defined quantum dot (QD). While these experimental findings are intriguing, their further implementation in nanoelectronic devices hinges upon the knowledge of key underpinning parameters, which still remain elusive. In this paper, we first summarize the experimental measurements of the deformation of graphene membranes due to interactions with the STM probe tip and a back gate electrode. We then carry out systematic coarse grained, (CG), simulations to offer a mechanistic interpretation of STM tip-induced straining of the graphene drumhead. Our findings reveal the effect of (i) the position of the STM probe tip relative to the graphene drumhead center, (ii) the sizes of both the STM probe tip and graphene drumhead, as well as (iii) the applied back-gate voltage, on the induced strain field and corresponding pseudomagnetic field. These results can offer quantitative guidance for future design and implementation of reversible and on-demand formation of graphene QDs in nanoelectronics., Comment: 21 pages, 9 figures

Designing high-finesse resonant cavities for electronic waves faces challenges due to short electron coherence lengths in solids. Previous approaches, e.g. the seminal nanometer-sized quantum corrals, depend on careful positioning of adatoms at clean surfaces. Here we demonstrate an entirely different approach, inspired by the peculiar acoustic phenomena in whispering galleries. Taking advantage of graphene's unique properties, namely gate-tunable light-like carriers, we create Whispering Gallery Mode (WGM) resonators defined by circular pn-junctions, induced by a scanning tunneling probe. We can tune the resonator size and the carrier concentration under the probe in a back-gated graphene device over a wide range, independently and in situ. The confined modes, revealed through characteristic resonances in the tunneling spectrum, originate from Klein scattering at pn junction boundaries. The WGM-type confinement and resonances are a new addition to the quantum electron-optics toolbox, paving the way to real-world electronic lenses and resonators.

Electron beam induced current (EBIC) is a powerful technique which measures the charge collection efficiency of photovoltaics with sub-micron spatial resolution. The exciting electron beam results in a high generation rate density of electron-hole pairs, which may drive the system into nonlinear regimes. An analytic model is presented which describes the EBIC response when the {\it total} electron-hole pair generation rate exceeds the rate at which carriers are extracted by the photovoltaic cell, and charge accumulation and screening occur. The model provides a simple estimate of the onset of the high injection regime in terms of the material resistivity and thickness, and provides a straightforward way to predict the EBIC lineshape in the high injection regime. The model is verified by comparing its predictions to numerical simulations in 1 and 2 dimensions. Features of the experimental data, such as the magnitude and position of maximum collection efficiency versus electron beam current, are consistent with the 3 dimensional model., Comment: 10 pages, 8 figures

Electron beam induced current (EBIC) is a powerful characterization technique which offers the high spatial resolution needed to study polycrystalline solar cells. Ideally, an EBIC measurement reflects the spatially resolved quantum efficiency of the device. In this work, a model for EBIC measurements is presented which applies when recombination within the depletion region is substantial. This model is motivated by cross-sectional EBIC experiments on CdS-CdTe photovoltaic cells which show that the maximum efficiency of carrier collection is less than 100 \% and varies throughout the depletion region. The model can reproduce experimental results only if the mobility-lifetime product $\mu\tau$ is spatially varying within the depletion region. The reduced collection efficiency is speculated to be related to high-injection effects, and the resulting increased radiative recombination., Comment: 10 pages, 7 figures

When graphene is close to charge neutrality, its energy landscape is highly inhomogeneous, forming a sea of electron-like and hole-like puddles, which determine the properties of graphene at low carrier density. However, the details of the puddle formation have remained elusive. We demonstrate numerically that in sharp contrast to monolayer graphene, the normalized autocorrelation function for the puddle landscape in bilayer graphene depends only on the distance between the graphene and the source of the long-ranged impurity potential. By comparing with available experimental data, we find quantitative evidence for the implied differences in scanning tunneling microscopy measurements of electron and hole puddles for monolayer and bilayer graphene in nominally the same disorder potential., Comment: 7 pages, 6 figures

Bilayer graphene has drawn significant attention due to the opening of a band gap in its low energy electronic spectrum, which offers a promising route to electronic applications. The gap can be either tunable through an external electric field or spontaneously formed through an interaction-induced symmetry breaking. Our scanning tunneling measurements reveal the microscopic nature of the bilayer gap to be very different from what is observed in previous macroscopic measurements or expected from current theoretical models. The potential difference between the layers, which is proportional to charge imbalance and determines the gap value, shows strong dependence on the disorder potential, varying spatially in both magnitude and sign on a microscopic level. Furthermore, the gap does not vanish at small charge densities. Additional interaction-induced effects are observed in a magnetic field with the opening of a subgap when the zero orbital Landau level is placed at the Fermi energy.

Graphene is a unique two-dimensional material with rich new physics and great promise for applications in electronic devices. Physical phenomena such as the half-integer quantum Hall effect and high carrier mobility are critically dependent on interactions with impurities/substrates and localization of Dirac fermions in realistic devices. We microscopically study these interactions using scanning tunneling spectroscopy (STS) of exfoliated graphene on a SiO2 substrate in an applied magnetic field. The magnetic field strongly affects the electronic behavior of the graphene; the states condense into welldefined Landau levels with a dramatic change in the character of localization. In zero magnetic field, we detect weakly localized states created by the substrate induced disorder potential. In strong magnetic field, the two-dimensional electron gas breaks into a network of interacting quantum dots formed at the potential hills and valleys of the disorder potential. Our results demonstrate how graphene properties are perturbed by the disorder potential; a finding that is essential for both the physics and applications of graphene., Comment: to be published in Nature Physics

Transport properties of metal-molecule-metal junctions containing monolayer of conjugated and saturated molecules with characteristic dimensions in the range of 30-300 nm are correlated with microscopic topography, stress and chemical bonding at metal-molecule interfaces. Our statistically significant dataset allows us to conclude that the conductivity of organic molecules ~1.5 nm long is at least 4 orders of magnitude lower than is commonly believed., Comment: 12 pages, 4 figures