Electronic transport in graphene with out-of-plane disorder

Graphene, a prototypical two-dimensional material, exhibits soft flexural modes that enable low-energy out-of-plane deformations (ripples, wrinkles, folds). Such deformations arise from growth and transfer processes (e.g., CVD cooling-induced substrate contraction, transfer steps) and from the interplay between bending rigidity and interlayer adhesion. Out-of-plane disorder impacts electronic structure and transport via curvature-induced pseudo-gauge fields, interlayer tunneling in collapsed regions, and local charge accumulation acting as scattering centers. Prior studies aimed to eliminate wrinkles or leverage controlled folding for device engineering, yet a detailed understanding of how different types of out-of-plane disorder and their commensuration affect ballistic transport remained incomplete. This work investigates the research question: how do commensuration, width, and morphology (wrinkles vs folds) of out-of-plane disorder control ballistic charge-carrier transmission in graphene? The study aims to clarify the mechanisms (e.g., interlayer tunneling and interference) and provide guidelines to control transport in realistic devices.

The literature establishes that graphene’s out-of-plane morphology strongly affects electronic properties. CVD-grown graphene often forms wrinkles and folds due to thermal contraction; transfer processes can also introduce disorder. Strategies to minimize wrinkles include matching substrate thermal expansion, strain engineering, and temperature control. Prior experimental and theoretical studies reported that curvature induces pseudo-gauge fields, collapsed regions enable interlayer tunneling, and local charges act as scattering centers affecting device performance. Controlled folding has been proposed to engineer charge-carrier dynamics. Stacking configurations (Bernal AB/BA and alternatives) and twisted bilayers/trilayers influence electronic structure, with known relaxation effects at small twist angles leading to domain formation and flat/remote band separation. However, a systematic, first-principles treatment of ballistic transport across relaxed wrinkle and fold structures, including the role of commensuration and interference, has been lacking.

• Structure generation and relaxation: Atomistic models of wrinkles/folds were constructed by imposing a compressive displacement Δw along crystallographic vectors v=(a,b), yielding periodic structures along v. Classical force-field relaxation using LAMMPS was employed, combining a bond-order potential for covalent bonding and a modified Kolmogorov–Crespi registry-dependent potential for interlayer van der Waals interactions. Energy minimization used conjugate-gradient and FIRE algorithms.
• Systems studied: Commensurate wrinkles (zigzag v=(1,0) and armchair v=(1,1)) yielding Bernal-like stacking in collapsed regions; incommensurate wrinkles along general (a,b) directions producing locally twisted bilayer regions characterized by twist angle θ and mini-Brillouin zone projections (class Ia: a⊥b mod 3 = 0; class Ib: otherwise). Folds (trilayer collapsed regions) were also modeled, including commensurate (zigzag) and incommensurate (e.g., v=(1,2)).
• Electronic structure and transport calculations: Ballistic transmission T(E,k||) computed using DFT+NEGF (TranSIESTA) and tight-binding (TB) Slater–Koster models. DFT used SIESTA/TranSIESTA with double-ζ plus polarization basis, LDA (Perdew–Zunger), basis energy shift 275 meV, real-space cutoff 250 Ry. Lead–device coupling and self-energies obtained via recursive Green’s functions; conductance computed as G(E)=Σk|| 2G0 T(k||,E) with G0=2e^2/h.
• Tight-binding details: Slater–Koster parameterization including curvature-dependent π (intralayer) and σ (interlayer) couplings using Vπ=2.7 eV, Vσ=0.48 eV, a0=1.42 Å, d0=3.35 Å, decay length r0=0.184a (a=√3 ag=2.46 Å), and local normal vectors for angle-dependent overlaps.
• Atomic chain model: A 1D nearest-neighbor TB chain with additional hopping t′ (t′/t≈0.18) modeling interlayer tunneling; parameters include ΔW (effective path difference) and l (loop length without interlayer hopping). Transmission analyzed via NEGF with first-order correction to Green’s function G0ΔhG0 to reveal interference-induced oscillation periodicity scaling with ΔW.
• Additional analyses: Momentum conservation arguments in incommensurate systems using mirror symmetry of stacking (Mx(kx,ky)=(−kx,ky)) to identify regions of suppressed backscattering; estimation of direct top–bottom layer coupling in folds via localized basis Hamiltonian (sisl) and Slater–Koster (found negligible, ~10^−4 eV).

• Commensurate wrinkles (zigzag and armchair with Bernal-like stacking):
  • Pronounced electron–hole asymmetry in transmission, linked to sublattice-specific interlayer coupling in collapsed regions (AB/AB′ stacking). Armchair wrinkles exhibit symmetry evolution with shear; near-Bernal armchair configurations show clearer asymmetry.
  • Strong, energy-wide transmission oscillations with peak spacing Δε proportional to 1/Δw. Oscillations are observed across k||, including at projected Dirac points.
  • Physical origin: quantum interference between intra- and interlayer transport channels. A minimal 1D chain model with additional hopping t′ reproduces oscillations, with period governed by the largest path difference ΔW; parameter l (loop region without interlayer hopping) weakly affects peak positions.
• Incommensurate (twisted) wrinkles:
  • Momentum conservation and mirror symmetry imply suppressed backscattering at the Dirac point where projected Dirac cones do not overlap. First-order correction G0ΔhG0 vanishes when k-states do not match, yielding transmission close to pristine graphene.
  • Near charge neutrality, transmission magnitudes follow Dirac cone projections: T≈2 for class Ia and T≈1 for class Ib configurations, indicating minimal interlayer tunneling.
  • At higher energies where the projected Dirac cones overlap (e.g., around E≈2 eV in a class Ib case), interlayer coupling induces transmission dips.
• Folds (trilayer collapsed regions):
  • Incommensurate folds exhibit enhanced backscattering compared to wrinkles due to the doubled interlayer tunneling channels, not due to direct top–bottom coupling (estimated ~10^−4 eV and negligible).
  • Quantitatively, for v=(1,2): average transmission in (−0.15 eV, 0.15 eV) is 0.727 for a fold of width l=40 Å, versus 0.908 for an incommensurate wrinkle with equivalent Δw=80 Å.
  • Commensurate zigzag folds show stronger backscattering and lower transmission than equivalent zigzag wrinkles.
• Overall: Lattice commensuration is the key control parameter for backscattering and interference-induced oscillations; incommensuration suppresses interlayer tunneling near neutrality, preserving near-pristine transport.

The study delineates how out-of-plane disorder affects ballistic transport by separating the roles of commensuration and geometry. In commensurate wrinkles, interlayer registry enables strong interlayer tunneling that interferes with intralayer paths, producing robust transmission oscillations whose period scales inversely with the wrinkle width. This directly addresses the research question by identifying interference between transport channels as the mechanism behind oscillatory conductance and electron–hole asymmetry. In contrast, incommensurate (twisted) wrinkles conserve momentum in a way that suppresses first-order backscattering at the Dirac point; thus, near the Fermi level, transmission resembles that of pristine graphene. When projected Dirac cones overlap at higher energies, backscattering re-emerges, explaining energy-dependent dips. Folds enhance backscattering relative to wrinkles primarily because the number of interlayer tunneling interfaces is doubled in the trilayer collapsed region. These insights emphasize commensuration as a decisive factor for tuning transport and suggest that controlling wrinkle orientation, width, and stacking can tailor device performance. The implications extend to mesoscopic transport regimes (diffusive/localized), where multiple scattering events will integrate these single-scattering properties, and to general 2D materials exhibiting similar out-of-plane disorder and local twisted interfaces.

This work establishes a comprehensive, first-principles picture of ballistic transport across graphene wrinkles and folds with fully relaxed lattices. Key contributions include: (i) identification of strong transmission oscillations in commensurate wrinkles arising from intra-/interlayer interference with Δε∝1/Δw; (ii) demonstration that incommensuration suppresses interlayer tunneling and backscattering near charge neutrality, preserving near-pristine transmission; and (iii) clarification that folds exhibit enhanced backscattering due to doubled interlayer channels rather than direct outer-layer coupling. These findings provide design guidelines for managing the impact of ubiquitous out-of-plane disorder in graphene devices via control of orientation, width, and stacking registry. Future research directions include: exploring small-twist-angle wrinkles/folds where relaxation induces AB/BA domains and transport gaps; bridging from ballistic to mesoscopic regimes; and extending the framework to other 2D materials with analogous out-of-plane disorder and interlayer coupling phenomena.

• The analysis focuses on single-event ballistic transmission across periodic wrinkles/folds; real devices may involve multiple, disordered defects and inelastic processes not explicitly modeled.
• DFT+NEGF calculations are computationally demanding for very small twist angles; such regimes are discussed qualitatively due to standard DFT limitations.
• Finite-size, periodic models assume ideal periodicity along the wrinkle direction and do not include temperature, phonon scattering, impurities, or substrate effects beyond structural relaxation inputs.
• Conversion from transmission to measurable conductance requires device width W; results are presented per transmission channel and as T(E,k||) maps.
• Electronic structure approximations (LDA, localized basis sets) and TB parameterizations may introduce quantitative uncertainties, though trends are robust.