Create Graphene & Carbon Nanotubes in Atomistix ToolKit

A theoretical framework for describing electronic transport in systems out of equilibrium using non-equilibrium Green's functions. The approach defines current operators as traces of operators describing incoming and outgoing electron flow, with couplings to electrodes described by self-energies. In the non-interacting case, this reduces to a transmission coefficient that depends on bias. The current expression in stationary state shows that current flowing into one electrode equals the opposite of current flowing out of the other, allowing the current to be expressed in terms of transmission coefficients and density of states. This framework combines transmission approach and rate equation approach into a unified formalism for nanoscopic devices.

Non-equilibrium Green's function (NEGF) formalism provides the theoretical foundation for quantum transport calculations in nanoscale devices. This approach constructs Green's functions from density functional Hamiltonians and computes current using Landauer's formula, which integrates transmission functions across occupied states in one electrode and unoccupied states in another. The transmission function characterizes the probability for electrons to pass between electrodes at given energies. NEGF enables modeling of quantum effects critical in modern nanoelectronics: tunneling currents when insulating layers become thin, quantum confinement causing state quantization, and leakage currents. Applications span knowledge devices, hybrid geometries (graphene-carbon nanotube junctions), defective systems, quantum dots, corrosion processes, molecular electronics, and DNA-based sensors.

NEGF provides rigorous quantum transport framework using Schrödinger equation for mechanics and self-energies (Σ₀, Σ₁, Σ₂) for scattering. Coherent transport shows quantum interference oscillations from standing waves. Momentum relaxation eliminates oscillations but preserves resistance slopes; phase relaxation eliminates oscillations without creating resistance. Resistance fundamentally stems from momentum relaxation, not energy dissipation—electrons can change momentum direction without energy loss. This challenges the paradigm that resistance requires dissipation.

The Non-Equilibrium Green Function (NEGF) methodology provides a comprehensive framework for describing electron transport through nanoscale devices by combining quantum mechanical principles (Schrödinger equation) with entropy-driven processes, enabling the calculation of current flow through contacts via self-energy terms (σ₁, σ₂) and scattering processes (σ₀), which can be applied to both coherent transport (ignoring channel interactions) and non-coherent transport (including electron-electron interactions and inelastic scattering), with applications ranging from quantized conductance in ballistic conductors to Hall effect phenomena and Coulomb blockade effects.

The Non-Equilibrium Green's Function (NEGF) method is a quantum transport formalism that extends the Schrödinger equation by incorporating self-energies (Sigma) to account for contact coupling and scattering processes, enabling the calculation of quantum interference effects and level broadening that cannot be captured by classical transport theories or the standard Schrödinger equation alone.

Carbon nanotubes can form defects during synthesis, including pentagons, heptagons, and Stone-Wales defects. These defects disrupt the perfect hexagonal carbon network and affect electronic properties. Carbon nanotubes exhibit either metallic or semiconducting behavior depending on their chirality (the way the graphene sheet is rolled). This property is determined by the nanotube's diameter and the angle of rolling. Metallic nanotubes have zero band gap, while semiconducting nanotubes have finite band gaps. The band gap of carbon materials depends on their size and structure. As the size of carbon nanostructures decreases, the band gap typically increases due to quantum confinement effects. This relationship is important for designing carbon materials with specific electronic properties for applications in electronics and optoelectronics.

The Stone-Wales defect is a 90° rotation of two bonded carbon atoms that changes their connectivity in carbon nanotubes and graphene, significantly affecting their electronic, chemical, and mechanical properties; this defect can be modeled computationally using simulation software like Device Studio by importing carbon atoms, building the nanotube structure, selecting two adjacent atoms, applying a 90° rotation, recalculating bonds, and performing SCF calculations followed by analysis of density of states, band structure, and other electronic properties.

Graphene defects include Stone-Wales defects (bond rotations creating pentagon-heptagon pairs) and vacancy defects (missing atoms creating pentagon-octagon configurations). Stone-Wales defects are unstable and disappear within seconds as bonds rotate back. Vacancy defects persist as absorbed atoms fill vacancies over time. Both defect types create local strain fields affecting electronic properties. The overall Burgers vector around Stone-Wales defects is zero, indicating local rearrangements without net lattice displacement. These defects are created by electron irradiation and evolve dynamically under observation.

Quantum ATK enables defect engineering studies: Carbon nanotube creation from graphene sheets using tube wrapper plugin; Defect introduction through geometry manipulation (rotation of atoms to create Stone-Wales defects); Structural visualization and verification; Conversion to device configuration for transport analysis; This capability enables study of defect effects on electronic and transport properties of carbon nanotube-based devices.

This video demonstrates how Stone-Wales defects in graphene nanoribbons can be modeled using the tight binding method in MATLAB, where the Hamiltonian matrix is generated from alpha (self-interaction) and beta (inter-atom interaction) matrices; the simulation shows that introducing a Stone-Wales defect at the symmetrical center of an armchair graphene nanoribbon increases its band gap compared to the pristine structure, as evidenced by the band structure plots and density of state calculations.

CNTFETs replace bulk silicon with carbon nanotubes as channel materials, addressing Moore's law scaling limits at sub-22nm where electron tunneling, leakage currents, and short channel effects emerge. The chiral vector Ĉh = nâ1 + mâ2 defines nanotube structure, with diameter dt = (√3 × aCC × √(m² + mn + n²))/π and chiral angle θ = tan⁻¹[(√3 × n)/(2m + n)]. Nanotubes are metallic when n=m, semi-metallic when n-m=3i, and semiconducting otherwise, with bandgap Eg = γ(2a/dt). Four main architectures exist: back-gated devices use randomly deposited CNTs on pre-patterned metal strips but suffer poor contacts; top-gated devices employ solution deposition, AFM/SEM localization, e-beam lithography, and thin gate dielectrics for superior performance; wrap-around gate CNTFETs (2008) gate the entire circumference via ALD-wrapped structures; suspended CNTFETs reduce substrate scattering but face mechanical and dielectric challenges. Material selection favors semiconducting SWCNTs with palladium contacts matching work functions.

A Carbon Nanotube Field Effect Transistor (CNTFET) is a nanoscale electronic device that uses a single-walled carbon nanotube as the conducting channel between source and drain electrodes, with a gate electrode wrapped around the nanotube to provide excellent electrostatic control; unlike silicon transistors at nanoscale dimensions which face issues with fabrication difficulty and poor carrier mobility, CNTFETs offer superior electrical properties including long mean free path, high carrier mobility, and improved electrostatics, making them promising candidates for replacing silicon in future nanoelectronic applications.

Carbon nanotube field effect transistors (CNTFETs) require specialized compact models due to challenges like metallic tube contamination. The SIKAN model developed for CNTFETs includes intrinsic and extrinsic parts, with metallic tubes represented in the extrinsic network. The model's modularity allows adaptation as technology improves—when metallic tubes become less prevalent, the extrinsic part can be eliminated while retaining the intrinsic semiconductor portion. Recent improvements added enhanced charge definition, better describing cutoff frequency performance. These models enable circuit-level performance projections, such as designing FSK and PSK modulators based on CNTFETs, demonstrating how compact models facilitate complex circuit design for emerging technologies.

Carbon nanotube field effect transistors (CNTFETs) are emerging nanoelectronic devices that address the limitations of silicon transistors at smaller scales, offering higher channel mobility, lower threshold voltage, and negligible leakage current; they are classified into bottom gate, top gate, and wraparound gate structures based on geometry, and into Schottky barrier and MOSFET-like types based on operation, with MOSFET-like CNTFETs being particularly promising for high-speed CMOS applications due to their superior electrostatic control and compatibility with conventional circuit designs.

Carbon nanotube field effect transistors (CNTFETs) are transistors fabricated with carbon nanotubes as the channel, placed between source and drain electrodes with a gate electrode controlling current flow; these devices offer advantages over silicon MOSFETs including lower switching delays, high electron mobility, quantum confinement effects, small size, faster operations, high gain, large on/off ratio, and room temperature stability, making them promising for next-generation logic circuits.