Electronic transport in graphene with out-of-plane disorder

Graphene, the first and most studied two-dimensional (2D) material, holds immense promise for technological applications. Its 2D nature leads to out-of-plane disorder (ripples, wrinkles, folds) stemming from the interplay between its self-adhesion and bending rigidity. These defects significantly impact graphene's electronic structure and transport properties. Curvature creates pseudo-gauge fields, while collapsed regions allow interlayer tunneling and act as scattering centers. Out-of-plane disorder arises during growth (e.g., chemical vapor deposition, CVD) due to substrate thermal contraction or during transfer processes. Various methods are employed to minimize wrinkles, such as using substrates with matched thermal expansion coefficients, strain engineering, and refined temperature protocols. However, despite experimental studies, a comprehensive understanding of the effects of this ubiquitous disorder on electronic transport is crucial for future applications. This work systematically investigates the electronic transport across graphene wrinkles and folds using first-principles calculations.

Previous research has explored the impact of out-of-plane disorder on graphene's electronic properties. Studies have shown that wrinkles and folds introduce pseudo-gauge fields due to curvature and create pathways for interlayer electron tunneling. The accumulation of charges in these disordered regions acts as scattering centers, affecting the performance of graphene-based electronic devices. Efforts have been made to minimize or eliminate wrinkles during graphene synthesis and transfer processes, and controlled folding has been proposed as a means to engineer charge-carrier dynamics. However, a detailed understanding of how different types of out-of-plane defects affect electronic transport remains an active area of research. This study builds upon prior work by employing first-principles computations to examine the effect of both commensurate and incommensurate wrinkles and folds on ballistic electron transmission.

Atomistic models of graphene with out-of-plane defects were created by applying compressive displacement (Δw) along specific crystallographic vectors (v). The resulting structures (ripples, wrinkles, folds) were relaxed using classical force-field simulations. Ballistic charge-carrier transmission was calculated using density functional theory (DFT) and the non-equilibrium Green's function (NEGF) formalism implemented in the TranSIESTA package. Tight-binding (TB) calculations using the Slater-Koster formalism were also performed for comparison. The DFT calculations employed a double-ζ plus polarization basis set with the local density approximation exchange-correlation functional. The TB model incorporated both interlayer and intralayer couplings. Transmission calculations used the NEGF method to determine the transmission probability as a function of energy and momentum parallel to the wrinkle. A simple one-dimensional tight-binding model was developed to investigate the mechanism underlying conductance oscillations observed in commensurate wrinkles, treating interlayer coupling as a perturbative correction to the transmission using a NEGF approach. The effect of wrinkle direction on transmission was analyzed by considering the projection of the 2D Brillouin zone onto the direction parallel to the wrinkle. The study distinguished between commensurate (Bernal stacking) and incommensurate (twisted bilayer) wrinkles and folds, analyzing the transmission characteristics in each case.

For commensurate zigzag and armchair wrinkles, the study observed pronounced electron-hole asymmetry and transmission oscillations across a wide energy range. These oscillations, found to be consistent in both DFT and TB calculations, are attributed to quantum interference between intra- and interlayer transport channels. The oscillation period was found to be inversely proportional to the wrinkle width (Δw). A simple one-dimensional tight-binding model accurately captures these oscillations, demonstrating that the interference path length (Δw) primarily determines the oscillation peaks. In contrast, incommensurate wrinkles showed significantly suppressed backscattering, and their transmission properties closely resembled those of flat graphene, indicating that the interlayer coupling is weak. In incommensurate folds, enhanced backscattering was observed compared to wrinkles, attributed to the doubled number of interlayer tunneling channels. However, direct coupling between the outermost layers in the folds was negligible. Commensurate zigzag folds showed strong backscattering, similar to commensurate wrinkles. For incommensurate wrinkles, the conservation of crystallographic orientation in the graphene sheets on either side of the wrinkle leads to suppressed backscattering near the Dirac point, with significant backscattering only occurring in regions of Dirac cone overlap at higher energies. The results indicate that the degree of backscattering is related to the commensuration of the layers; commensurate structures show significant oscillations and backscattering, whereas incommensurate structures exhibit significantly reduced backscattering.

This research provides valuable insights into the effect of ubiquitous out-of-plane disorder on graphene's electronic transport. The findings demonstrate the crucial role of interlayer coupling and lattice commensuration in determining the transmission characteristics. The observed oscillations in commensurate wrinkles highlight the importance of quantum interference effects, while the suppression of backscattering in incommensurate wrinkles suggests strategies for designing devices with minimal scattering. The enhanced backscattering in folds compared to wrinkles provides additional understanding of the interplay between defect structure and transport. While the study focuses on ballistic transmission across individual defects, the results can inform predictions about mesoscopic transport properties resulting from multiple scattering events. This work opens avenues for designing graphene-based devices by exploiting the interplay between interference phenomena and electromechanical coupling, building on previous work demonstrating electromechanical oscillations in bilayer graphene. The principles established here are expected to apply to other 2D materials.

This study comprehensively investigated the electronic transport across graphene wrinkles and folds, revealing the significant impact of interlayer coupling and commensuration. Commensurate wrinkles exhibit transmission oscillations due to quantum interference, while incommensurate wrinkles show minimal backscattering. Folds show enhanced backscattering due to increased interlayer contact. These findings offer valuable insights for designing and optimizing graphene-based electronic devices and suggest future research directions focusing on electromechanical coupling in these systems and the extension of these principles to other 2D materials, particularly exploring the small-angle twist regime.

The study primarily focuses on ballistic transport across individual defects and does not directly address the effects of multiple scattering or other sources of disorder. The models considered are periodic representations of wrinkles and folds, which might not perfectly capture the complex and often irregular nature of these defects in real samples. The DFT calculations were performed within the local density approximation, which may introduce some limitations in the accuracy of the electronic structure calculations, particularly for systems with strong electronic correlations. Future work could explore non-periodic defects and use more advanced exchange-correlation functionals to refine the accuracy of the calculations.