Graphene Defects Exhibit Intermediate Diffusive-ballistic Electron Conduction, Underestimating Electrochemical Potential with Diffusive Models

The behaviour of electrons in nanoscale materials presents a fundamental challenge to conventional understanding of conductivity, particularly when features approach the size of an electron’s mean free path. Toni Markovic, Wei Huang, William S. Huxter, and colleagues at ETH Zurich now demonstrate a previously unobserved intermediate regime of electron conduction around defects in graphene. Their research combines nanoscale topography measurements with local electrochemical potential mapping, revealing that electron transport around these defects is neither purely diffusive nor entirely ballistic, but exists in a state between the two. This discovery significantly improves understanding of electron behaviour in nanoscale materials, showing that ballistic contributions become noticeable even at mesoscopic length scales and are substantially underestimated by existing diffusive models.

Graphene’s Electronic Transport and Defect Influence

A comprehensive study of graphene’s electronic properties reveals how electrons move through the material and how imperfections affect this movement. Researchers investigated ballistic transport, where electrons travel without scattering, and the impact of defects on conductivity, utilizing scanning tunneling potentiometry to visualize these effects. The study also explored electron behavior at graphene edges and contacts with other metals. The research highlights graphene creation, particularly on silicon carbide substrates, and how the growth process influences material quality and defects. Scientists investigated the possibility that electrons in graphene can behave like a fluid, linked to interactions between electrons and resulting in unusual transport phenomena.

Detailed analysis using scanning tunneling potentiometry mapped local electrical potential and resistance, providing insights into defects and edge effects. Foundational work established the principles of residual resistivity and the impact of localized scattering on electron transport. Recent experiments directly observed current whirlpools in graphene at room temperature, providing strong evidence for the viscous fluid behavior of electrons. Computational modeling using the Lattice Boltzmann Method provided a theoretical framework for understanding experimental observations of electron fluid behavior. This body of work represents a vibrant and interdisciplinary field, combining materials science, condensed matter physics, and advanced experimental techniques. The focus on electron fluid behavior is particularly exciting, suggesting that graphene may exhibit novel electronic properties not found in conventional materials.

Defect Potential Mapping via Mesoscopic Simulation

Scientists investigated charge transport around nanoscale defects in materials, simultaneously resolving topographical features and local electrochemical potential using scanning tunneling potentiometry. The team focused on pits within the material to probe both diffusive and ballistic contributions to electron scattering. Experiments revealed transport behavior falling between purely diffusive and ballistic limits, demonstrating that standard diffusive models underestimate the electrochemical potential around the defects. To further explore this intermediate transport regime, researchers developed a mesoscopic transport model using the Lattice Boltzmann Method, a computational technique for simulating fluid dynamics.

This model iteratively solves for quasi-particle density functions, representing the material’s geometry and incorporating diffusive scattering by adding momentum relaxation during collision processes. By adjusting simulation parameters, scientists tuned the model from the diffusive to the ballistic regime. The simulations computed steady-state quasi-particle densities to determine current density and local electrochemical potential, generating maps of the electrochemical potential. By systematically varying pit radius and mean free path, the team compared simulated dipole magnitudes to experimental data.

Results demonstrated that increasing the mean free path led to dipole magnitudes exceeding the diffusive limit, indicating ballistic contributions. Analysis of the dipole magnitude as a function of pit radius revealed a decrease from a value of one, corresponding to purely diffusive transport. Scientists plotted the dipole magnitude normalized by pit radius against the ratio of pit radius to mean free path, finding that all simulated data aligned to a single curve, demonstrating an inverse proportionality and capturing the smooth transition into the intermediate regime. This scaling behavior mirrors observations of conductance in the intermediate regime, highlighting the consistency between simulation and experimental findings.

Non-Diffusive Charge Transport Around Graphene Pits

Researchers have achieved a detailed understanding of charge transport around nanoscale defects in graphene, revealing behavior that deviates from traditional diffusive models. The study focused on pits, nanoscale holes formed during graphene growth on silicon carbide, and employed scanning tunneling potentiometry to simultaneously map the surface topography and local electrochemical potential. Experiments conducted at both 295 K and 90 K demonstrate that the electrochemical potential around these pits is significantly underestimated by models assuming purely diffusive transport, indicating the presence of non-diffusive effects. Detailed measurements of pit topography reveal typical depths of 600 picometers, corresponding to the bilayer graphene height above the underlying buffer layer.

Scanning tunneling spectroscopy confirms the distinct electronic properties of monolayer graphene, bilayer graphene, and the pit regions, showing substantially less conductive behavior within the pits compared to the surrounding graphene. By applying a bias current and measuring the electrochemical potential, the team observed a clear dipolar structure around the pits, with increased potential on one side and decreased potential on the other, directly correlating with the pit edges. Quantitative analysis of the electrochemical potential profiles reveals that ballistic effects become important even at scatterer sizes significantly larger than the electron mean free path. Researchers developed both diffusive and ballistic models to analyze the observed potential profiles, finding that the ballistic model more accurately describes the experimental data. The ballistic model, based on the Landauer formula, predicts the potential around a circular scatterer, and the team found a strong correlation between the model predictions and the measured electrochemical potential. These findings demonstrate that geometric factors, specifically the shape of the scatterer, play a crucial role in determining the balance between ballistic and diffusive transport, even at mesoscopic length scales.

Graphene Pit Scattering, Ballistic to Diffusive Transition

This research demonstrates that naturally occurring pits in graphene significantly scatter electrons, exhibiting transport behavior between purely diffusive and ballistic regimes. Experiments reveal that the electrochemical potential around these defects is substantially larger than predicted by models assuming only diffusive transport, even when the pit dimensions exceed the electron mean free path. The magnitude of this scattering effect is strongly influenced by the shape of the pits. Notably, the team discovered that ballistic contributions to electron transport become prominent at feature sizes larger than the mean free path, increasing rapidly as the size decreases, and becoming relevant at mesoscopic length scales. This finding has important implications for nanoscale device applications, suggesting that electron transport in graphene is more complex than previously understood.