Local transport measurements at mesoscopic length scales using scanning tunneling potentiometry

Landauer Resistivity Dipole at One-Dimensional Defect Revealed via near-Field Photocurrent Nanoscopy

The fundamental question of how to describe ohmic resistance at the nanoscale was answered by Landauer in his seminal picture of the Landauer resistivity dipole (LRD). While this picture is theoretically well understood, experimental studies remain scarce due to the need for noninvasive local probes. Here, we use the nanometer lateral resolution of near-field photocurrent imaging to thoroughly characterize a monolayer-bilayer graphene interface. Via systematic tuning of charge carrier density and current flow, we detected charge carrier accumulation around this nearly ideal one-dimensional defect due to the formation of the LRDs. We found that, at low doping levels, the photocurrent exhibits the same polarity as the applied source-drain voltage, reflecting carrier concentration changes induced by the LRDs. This signature disappears at higher charge carrier densities in agreement with the numerical calculations performed. Photocurrent nanoscopy can thus serve as a noninvasive technique to study local dissipation at hidden interfaces.

Intermediate Diffusive-Ballistic Electron Conduction Around Mesoscopic Defects in Graphene

Nondiffusive effects in charge transport become relevant as device sizes and features become comparable to the electronic mean free path. As a model system, we investigated the electric transport around mesoscopic defects in graphene with scanning tunneling potentiometry. Diffusive and ballistic contributions to the scattering dipole are probed by simultaneously resolving the nanoscale topography of pits in the graphene layer and measuring the local electrochemical potential in the surrounding area. We find evidence of transport in the intermediate regime between the diffusive and ballistic limits such that the magnitude of the electrochemical potential around the defects is substantially underestimated by diffusive models. Our experiments and modeling are supported by lattice-Boltzmann simulations, which highlight the importance of the ratio between defect size and mean free path in the intermediate transport regime. The magnitude of the scattering dipole depends on the shape of the pits in both the ballistic and diffusive transport modes. Remarkably, ballistic contributions to the electron transport are found at feature sizes larger than the mean free path and rapidly increase at lower sizes, having a noticeable impact already on mesoscopic length scales.

Imaging the breaking of electrostatic dams in graphene for ballistic and viscous fluids

The charge carriers in a material can, under special circumstances, behave as a viscous fluid. In this work, we investigated such behavior by using scanning tunneling potentiometry to probe the nanometer-scale flow of electron fluids in graphene as they pass through channels defined by smooth and tunable in-plane p-n junction barriers. We observed that as the sample temperature and channel widths are increased, the electron fluid flow undergoes a Knudsen-to-Gurzhi transition from the ballistic to the viscous regime characterized by a channel conductance that exceeds the ballistic limit, as well as suppressed charge accumulation against the barriers. Our results are well modeled by finite element simulations of two-dimensional viscous current flow, and they illustrate how Fermi liquid flow evolves with carrier density, channel width, and temperature.

Performance analysis and implementation of a scanning tunneling potentiometry setup: Toward low-noise and high-sensitivity measurements of the electrochemical potential

Scanning tunneling potentiometry allows for studying charge transport on the nanoscale to relate the local electrochemical potential to morphological features of thin films or two-dimensional materials. To resolve the influence of atomic-scale defects on the charge transport, sub-µV sensitivity for the electrochemical potential is required. Here, we present a complete analysis of the noise in scanning tunneling potentiometry for different modes of operation. We discuss the role of various noise sources in the measurements and technical issues for both dc and ac detection schemes. The influence of the feedback controller in the determination of the local electrochemical potential is taken into account. Furthermore, we present a software-based implementation of the potentiometry technique in both dc and ac modes in a commercial scanning tunneling microscopy setup with only the addition of a voltage-controlled current source. We directly compare the ac and dc modes on a model resistor circuit and on epitaxial graphene and draw conclusions on the advantages and disadvantages of each mode. The effects of sample heating and the occurrence of thermal voltages are discussed.

Non-Invasive Nanoscale Potentiometry and Ballistic Transport in Epigraphene Nanoribbons

The recent observation of non-classical electron transport regimes in two-dimensional materials has called for new high-resolution non-invasive techniques to locally probe electronic properties. We introduce a novel hybrid scanning probe technique to map the local resistance and electrochemical potential with nm- and μV resolution, and we apply it to study epigraphene nanoribbons grown on the sidewalls of SiC substrate steps. Remarkably, the potential drop is non-uniform along the ribbons, and μm-long segments show no potential variation with distance. The potential maps are in excellent agreement with measurements of the local resistance. This reveals ballistic transport, compatible with μm-long room-temperature electronic mean-free paths.

New imaging modes for analyzing suspended ultra-thin membranes by double-tip scanning probe microscopy

Scanning probe microscopy (SPM) techniques are amongst the most important and versatile experimental methods in surface- and nanoscience. Although their measurement principles on rigid surfaces are well understood and steady progress on the instrumentation has been made, SPM imaging on suspended, flexible membranes remains difficult to interpret. Due to the interaction between the SPM tip and the flexible membrane, morphological changes caused by the tip can lead to deformations of the membrane during scanning and hence significantly influence measurement results. On the other hand, gaining control over such modifications can allow to explore unknown physical properties and functionalities of such membranes. Here, we demonstrate new types of measurements that become possible with two SPM instruments (atomic force microscopy, AFM, and scanning tunneling microscopy, STM) that are situated on opposite sides of a suspended two-dimensional (2D) material membrane and thus allow to bring both SPM tips arbitrarily close to each other. One of the probes is held stationary on one point of the membrane, within the scan area of the other probe, while the other probe is scanned. This way new imaging modes can be obtained by recording a signal on the stationary probe as a function of the position of the other tip. The first example, which we term electrical cross-talk imaging (ECT), shows the possibility of performing electrical measurements across the membrane, potentially in combination with control over the forces applied to the membrane. Using ECT, we measure the deformation of the 2D membrane around the indentation from the AFM tip. In the second example, which we term mechanical cross-talk imaging (MCT), we disentangle the mechanical influence of a scanning probe tip (e.g. AFM) on a freestanding membrane by means of independently recording the response of the opposing tip. In this way we are able to separate the tip-induced membrane deformation topography from the (material-dependent) force between the tip and the membrane. Overall, the results indicate that probing simultaneously both surfaces of ultra-thin membranes, such as suspended 2D materials, could provide novel insights into the electronic properties of the materials.

Conductivity map from scanning tunneling potentiometry

We present a novel method for extracting two-dimensional (2D) conductivity profiles from large electrochemical potential datasets acquired by scanning tunneling potentiometry of a 2D conductor. The method consists of a data preprocessing procedure to reduce/eliminate noise and a numerical conductivity reconstruction. The preprocessing procedure employs an inverse consistent image registration method to align the forward and backward scans of the same line for each image line followed by a total variation (TV) based image restoration method to obtain a (nearly) noise-free potential from the aligned scans. The preprocessed potential is then used for numerical conductivity reconstruction, based on a TV model solved by accelerated alternating direction method of multiplier. The method is demonstrated on a measurement of the grain boundary of a monolayer graphene, yielding a nearly 10:1 ratio for the grain boundary resistivity over bulk resistivity.

Scanning tunneling potentiometry implemented into a multi-tip setup by software

We present a multi-tip scanning tunneling potentiometry technique that can be implemented into existing multi-tip scanning tunneling microscopes without installation of additional hardware. The resulting setup allows flexible in situ contacting of samples under UHV conditions and subsequent measurement of the sample topography and local electric potential with resolution down to Å and μV, respectively. The performance of the potentiometry feedback is demonstrated by thermovoltage measurements on the Ag/Si(111)-(√3×√3)R30° surface by resolving a standing wave pattern. Subsequently, the ability to map the local transport field as a result of a lateral current through the sample surface is shown on Ag/Si(111)-(√3×√3)R30° and Si(111) - (7 × 7) surfaces.

Spatial extent of a Landauer residual-resistivity dipole in graphene quantified by scanning tunnelling potentiometry

Electronic transport on a macroscopic scale is described by spatially averaged electric fields and scattering processes summarized in a reduced electron mobility. That this does not capture electronic transport on the atomic scale was realized by Landauer long ago. Local and non-local scattering processes need to be considered separately, the former leading to a voltage drop localized at a defect, the so-called Landauer residual-resistivity dipole. Lacking precise experimental data on the atomic scale, the spatial extent of the voltage drop remained an open question. Here, we provide an experimental study showing that the voltage drop at a monolayer-bilayer boundary in graphene clearly extends spatially up to a few nanometres into the bilayer and hence is not located strictly at the structural defect. Moreover, different scattering mechanisms can be disentangled. The matching of wave functions at either side of the junction is identified as the dominant process, a situation similar to that encountered when a molecule bridges two contacts.

A first principles scanning tunneling potentiometry study of an opaque graphene grain boundary in the ballistic transport regime

We report on a theoretical interpretation of scanning tunneling potentiometry (STP), formulated within the Keldysh non-equilibrium Green's function description of quantum transport. By treating the probe tip as an electron point source/sink, it is shown that this approach provides an intuitive bridge between existing theoretical interpretations of scanning tunneling microscopy and STP. We illustrate this through ballistic transport simulations of the potential drop across an opaque graphene grain boundary, where atomistic features are predicted that might be imaged through high resolution STP measurements. The relationship between the electrochemical potential profile measured and the electrostatic potential drop across such a nanoscale defect is also explored in this model system.