Single-layer graphene represents one of the most revolutionary materials discovered in modern science. Since its first isolation in 2004 by Andre Geim and Konstantin Novoselov—work that earned them the 2010 Nobel Prize in Physics—this remarkable two-dimensional material has captured the attention of researchers worldwide and spawned an entirely new field of materials science.

What makes single-layer graphene so extraordinary is not just one exceptional property, but rather an unprecedented combination of characteristics that seemed almost impossible before its discovery. It is simultaneously the thinnest material known to exist, yet stronger than steel; it conducts electricity better than copper while remaining completely transparent; and it exhibits quantum mechanical behaviors typically seen only in exotic physics experiments, all at room temperature.

This comprehensive guide explores the fundamental properties, structural characteristics, and potential applications of single-layer graphene, providing both scientific depth and practical understanding of why this material continues to drive innovation across multiple technological frontiers.

What Is Single-Layer Graphene (SLG)? Definition & Description

• A single-atom-thin material that can be exfoliated from graphite (pencil lead)
• Made entirely of carbon atoms arranged in a hexagonal pattern, similar to chicken wire
• The most microscopic shaving of pencil lead you can imagine
• Honeycomb lattice structure with lattice constant a₀ = 2.47 Å

Single-layer graphene (SLG) is a single-atom-thin material that can be exfoliated from the same graphite that is found in pencil lead. The ultrathin material is made entirely from carbon atoms that are arranged in a simple hexagonal pattern, similar to that of chicken wire. To put this in perspective, single-layer graphene represents the most microscopic shaving of pencil lead you can imagine – literally the thinnest possible layer that can be peeled away from bulk graphite.

Graphene is a two-dimensional (2D) form of carbon allotrope, essentially a monolayer of sp² bonded carbon atoms. The material forms a distinctive honeycomb lattice structure with precise atomic spacing. The lattice constant a₀ = 2.47 Å defines the fundamental geometric parameters of this crystalline structure, with the in-plane lattice constant of 2.47 Å serving as the basic unit of measurement for the hexagonal arrangement.

Since its isolation in 2004, single-layer graphene has revolutionized materials science by demonstrating that two-dimensional materials can exist as stable, free-standing structures with remarkable properties that differ dramatically from their three-dimensional bulk counterparts.

What Are The Physical Properties Of Single-Layer Graphene (SLG)?

• Thinnest material known in the world
• Incredibly light and flexible
• Hundreds of times stronger than steel
• More electrically conductive than copper
• 2D form of carbon allotrope with sp² bonded carbon atoms

Single-layer graphene possesses an extraordinary combination of physical properties that have earned it numerous superlatives in materials science. It’s hard to believe that a single material can be described by as many superlatives as graphene can.

First and foremost, graphene is not just the thinnest material known in the world, but also incredibly light and flexible, hundreds of times stronger than steel, and more electrically conductive than copper. This unique combination of properties stems from its atomic structure – a 2D form of carbon allotrope with sp² bonded carbon atoms.

The mechanical strength of graphene is particularly remarkable. Despite being only one atom thick, it demonstrates tensile strength that far exceeds conventional materials, making it one of the strongest materials ever tested. Its flexibility allows it to be bent and twisted without breaking, while maintaining its structural integrity.

From an electrical standpoint, graphene’s conductivity surpasses that of copper, which has traditionally been the gold standard for electrical conductors. This exceptional conductivity, combined with its transparency and flexibility, opens up possibilities for applications that were previously impossible with conventional materials.

Note: Some property values can vary depending on measurement conditions and sample quality, as factors such as substrate interactions, defects, and environmental conditions can significantly influence measured properties.

What Are Mechanical Properties Of Single-Layer Graphene (SLG)?

The mechanical properties of single-layer graphene have been extensively studied through both experimental and computational approaches, revealing fascinating characteristics that distinguish it from multilayer systems.

• Young’s modulus determination through uniaxial tension
• Elastic properties different from multilayer systems
• Fracture mechanics and failure mechanisms
• Thermal conductivity measurements and calculations
• Phonon dispersion relationships
• Heat capacity and thermal expansion

Stress-Strain Behavior

Stress-strain curves offer essential insight into the mechanical behavior of materials, particularly their elastic properties and failure mechanisms. For single-layer graphene, researchers have conducted detailed analyses using uniaxial tension along the armchair direction to determine key mechanical parameters.

Starting from fully relaxed atomic structures, with both atomic positions and cell parameters optimized, researchers apply uniaxial strain by incrementally increasing the deformation up to 20%. At each strain level, the structure is re-relaxed using optimization algorithms with a force convergence threshold of 0.001 eV/Å, while keeping the cell fixed along the transverse directions.

In the linear elastic regime, the slope of each curve yields the Young’s modulus, E=σ/ε, characterizing the stiffness of the material. Single-layer graphene demonstrates exceptional stiffness, with elastic properties that differ significantly from multilayer graphene systems. Studies have shown that the trilayer structure exhibits the highest Young’s modulus, indicating increased stiffness with additional layers compared to monolayer graphene.

Molecular Dynamics Simulations & Defect Behavior

Molecular dynamics simulations have provided detailed insights into the mechanical behavior of single-layer graphene under various conditions. All molecular dynamics simulations are conducted using a Langevin thermostat, targeting the desired temperature with a friction coefficient of 0.01 fs⁻¹ and 1fs time-step, allowing researchers to study the material’s response to thermal fluctuations and overall dynamical behavior.

The material’s response to defects has been extensively studied through geometry optimization experiments. Point defects including vacancies, Stone-Wales (SW) defects, intralayer interstitials play crucial roles in determining mechanical properties. Vacancy migration pathways have been calculated using the Nudged Elastic Band (NEB) method, revealing how atomic-scale defects influence the overall mechanical performance of the material.

Defect formation energies provide quantitative measures of material stability. For example, vacancy formation in graphene systems requires specific energy inputs, while Stone-Wales defects involve the rotation of carbon-carbon bonds without removing atoms from the lattice.

Thermal Properties

The thermal properties of single-layer graphene are equally impressive and have been studied through various computational and experimental methods. Thermal conductivity measurements reveal exceptionally high values, with experimental data indicating that as temperature increases, thermal conductivity significantly decreases due to enhanced phonon-phonon scattering at high temperatures.

Phonon dispersion relationships play a crucial role in determining thermal properties. The phonon spectrum of single-layer graphene shows characteristic features including acoustic and optical branches that contribute to its thermal transport properties. Thermal conductivity can be calculated using the Green-Kubo method based on the fluctuation-dissipation theorem: κ = V/(kBT²) ∫₀^∞ ⟨J(0),J(t)⟩ dt, where V is the system volume, kB is the Boltzmann constant, T is the temperature, J(t) is the heat flux at time t, and ⟨J(0),J(t)⟩ is the autocorrelation function of the heat flux.

Heat capacity and thermal expansion measurements provide additional insights into the material’s thermal behavior under different temperature conditions. The velocity auto-correlation function (VACF) provides insights into the temporal correlations of particle velocities in graphene, defined as Cv(t) = ⟨v(0)·v(t)⟩, where v(t) is the velocity vector of a particle at time t.

What Are The Electronic Properties Of Single-Layer Graphene (SLG)?

The electronic properties of single-layer graphene are perhaps its most remarkable and scientifically significant characteristics, setting it apart from all other known materials.

Its 2D nature results in charge carriers that behave as relativistic massless Dirac fermions, obeying a linear energy dispersion. This unique electronic behavior means that electrons in graphene move as if they have no mass and follow relativistic physics, despite being in a solid-state material at room temperature.

• Charge carriers behave as relativistic massless Dirac fermions
• Linear energy dispersion relationship
• Very high field mobility
• Anomalous quantum Hall effect
• Dirac points at K and K’ valleys in the Brillouin zone

Valley Charge Conservation & Electronic Structure

The low energy states of single-layer graphene are contributed by those from the atomic K and K’ valleys, which are approximately decoupled from each other at the non-interacting level for small twist angles when considering multilayer systems. This valley charge conservation is generally assumed for graphene systems, providing an additional degree of freedom for electronic applications.

The electronic structure can be described using a pz tight-binding (TB) Hamiltonian approach. The electronic properties are computed using tight-binding models where hopping energies tnm between carbon C-sites are determined by the standard Slater-Koster formula: tnm = cos² φnm Vppσ(rnm) + sin² φnm Vppπ(rnm), where the direction cosine of rnm along the z-axis is cos φnm = znm/rnm.

Dirac Fermion Hamiltonian & Band Structure Features

The Dirac fermion behavior near the K and K’ points can be described by the Hamiltonian hμ(k) = -ℏvF k·σμ, where vF is the Fermi velocity, k is the wave vector measured from the Dirac point, and σμ are Pauli matrices defined in the sublattice space.

Very high field mobility allows electrons to move through the material with minimal scattering, resulting in exceptional electrical conductivity. This high mobility is maintained even at room temperature, unlike many other high-mobility materials that require cryogenic conditions.

Van Hove Singularities & Quantum Effects

Van Hove singularities emerge in the density of states at specific energies, creating peaks that can significantly influence the electronic properties. These singularities occur when the density of electronic states becomes very large due to the specific geometry of the energy bands.

Anomalous quantum Hall effect occurs in single-layer graphene, where the quantum Hall plateaus appear at unusual filling factors compared to conventional two-dimensional electron systems. This effect provides fundamental insights into the relativistic nature of charge carriers in graphene.

The electronic band structure features Dirac points at K and K’ valleys in the Brillouin zone, where the conduction and valence bands meet at a single point, creating a zero-gap semiconductor behavior. These Dirac points are responsible for the unique linear dispersion relationship that gives rise to the massless Dirac fermion behavior.

What Are The Crystal Structure Properties Of Single-Layer Graphene (SLG)?

The crystal structure of single-layer graphene is fundamental to understanding its remarkable properties and serves as the foundation for more complex multilayer systems.

• Fundamental matrix: A = a₀(1, -1/2; 0, √3/2)
• Two carbon atoms per unit cell with positions
• Sublattice structure with A and B sublattices
• Dirac cones at K and K’ points
• Valley degeneracy (K and K’ valleys)
• Spin degeneracy
• Massless Dirac fermion behavior near charge neutrality point

Unit Cell & Geometric Parameters

The basic structural unit of single-layer graphene is defined by its fundamental matrix: A = a₀(1, -1/2; 0, √3/2), where a₀ represents the lattice constant. This mathematical description captures the hexagonal symmetry of the graphene lattice with precise geometric relationships.

Each unit cell of graphene has two carbon atoms with shifts bₐ=(0,0)ᵀ and bᵦ=a₀(0,√3/3)ᵀ, labeled “A site” and “B site”. This bipartite lattice structure creates a sublattice arrangement where carbon atoms can be classified into two distinct sublattices, A and B, which play crucial roles in the electronic properties of the material.

The lattice constant a₀ = 2.47 Å defines the fundamental spacing between carbon atoms in the hexagonal arrangement. While single-layer graphene has no interlayer spacing by definition, this parameter becomes crucial when considering its role as a building block for multilayer systems, where typical interlayer distances are approximately 3.35 Å.

Electronic Band Structure

The electronic band structure of single-layer graphene is characterized by several key features that distinguish it from conventional materials:

Dirac cones at K and K’ points represent the most distinctive feature of graphene’s band structure. At these special points in the Brillouin zone, the conduction and valence bands meet in a linear fashion, creating cone-shaped energy-momentum relationships. The Brillouin zone structure includes high-symmetry points Γ, K, and M, with the K and K’ points being particularly important for the low-energy electronic properties.

Valley degeneracy (K and K’ valleys) provides an additional degree of freedom for electrons, as they can occupy states in either of the two equivalent valleys. This valley degree of freedom has been exploited in proposed valleytronic devices and contributes to the overall electronic degeneracy of the system.

Spin degeneracy adds another factor of two to the total degeneracy, meaning that each energy level can accommodate electrons with both spin orientations. Combined with valley degeneracy, this results in a fourfold degeneracy for each energy level.

Near the charge neutrality point, massless Dirac fermion behavior emerges, where electrons behave as if they have zero effective mass and follow relativistic dispersion relations typically seen only in high-energy physics.

Role As Building Block

• SLG serves as the fundamental building block for multilayer graphene assemblies
• Weak van der Waals interactions between layers
• Can be stacked and twisted to create novel electronic properties

Single-layer graphene serves a fundamental role as the basic building block for more complex graphene-based structures and devices. SLG serves as the fundamental building block for multilayer graphene assemblies, providing the foundation upon which more sophisticated structures can be constructed.

The interactions between multiple graphene layers are governed by weak van der Waals interactions between layers. These relatively weak interlayer forces (compared to the strong covalent bonds within each layer) allow for the creation of multilayer structures while preserving many of the unique properties of individual graphene sheets. The van der Waals interactions are strong enough to hold layers together but weak enough to allow for relative motion and rotation between layers.

One of the most exciting developments in graphene research involves the ability to stack and manipulate multiple layers: Can be stacked and twisted to create novel electronic properties. When two or more graphene layers are stacked with specific relative orientations, entirely new electronic behaviors can emerge. For example, when two graphene layers are stacked atop each other at just the right angle, the layered structure morphs into an unconventional superconductor, allowing electric currents to pass through without resistance or wasted energy.

This “magic-angle” phenomenon has opened up the entire field of twistronics, where the electronic properties of layered materials can be precisely controlled through their relative twist angles. The capability to fabricate 2D architectures with specific combinations of stacking or layer orientations presents almost unlimited possibilities for devices with novel functionality emerging from the coupling of layer-specific elementary excitations.

Note: The “magic angle” reference refers to twisted bilayer graphene specifically, where the relative rotation between two graphene layers creates unique superconducting properties at approximately 1.1 degrees.

Applications

The unique combination of properties in single-layer graphene has led to numerous potential applications across various technological fields.

• Transparent electrodes due to high conductivity and transparency
• Flexible electronics applications
• Sensor applications exploiting high surface area

Electronic & Sensor Applications

Transparent electrodes due to high conductivity and transparency represent one of the most commercially viable applications for single-layer graphene. The material’s exceptional electrical conductivity, combined with its optical transparency, makes it an ideal candidate for replacing traditional transparent conductors like indium tin oxide in displays, solar cells, and touchscreens.

Flexible electronics applications leverage graphene’s mechanical flexibility and electrical properties. The material can be bent, twisted, and stretched while maintaining its conductivity, enabling the development of flexible displays, wearable electronics, and conformable sensors that can adapt to curved surfaces.

Sensor applications exploiting high surface area take advantage of the fact that every atom in single-layer graphene is a surface atom, providing maximum exposure to the environment. This high surface-to-volume ratio, combined with graphene’s sensitivity to environmental changes, makes it excellent for detecting gases, biological molecules, and other chemical species at very low concentrations. The material’s electronic properties change measurably in response to adsorbed molecules, enabling highly sensitive detection systems.

Energy Storage & Quantum Applications

Energy storage applications benefit from graphene’s high surface area, excellent electrical conductivity, and chemical stability. Single-layer graphene can serve as electrode material in batteries and supercapacitors, where its high surface area provides numerous sites for charge storage and its conductivity ensures efficient charge transport.

Quantum computing applications represent one of the most promising frontiers for single-layer graphene. The material’s unique electronic properties, including its Dirac fermion behavior and valley degrees of freedom, make it an attractive platform for quantum information processing. The ability to control electronic states through external fields and the material’s coherence properties are particularly valuable for quantum device applications.

Advanced Materials & Composites

Composite materials represent another significant application area where single-layer graphene can dramatically improve the properties of conventional materials. When incorporated into polymers, metals, or ceramics, even small amounts of graphene can significantly enhance mechanical strength, electrical conductivity, and thermal properties of the composite.

The material’s exceptional strength-to-weight ratio makes it particularly valuable for aerospace and automotive applications where weight reduction without compromising structural integrity is crucial. Additionally, graphene’s thermal conductivity can help address heat management challenges in electronic devices and industrial applications.

The exceptional properties of single-layer graphene continue to drive research into new applications, from advanced sensors and energy devices to next-generation electronic components, positioning it as one of the most promising materials for future technological innovations.

Final Thoughts

From transition metal dichalcogenides to hexagonal boron nitride, and from quantum computers that could revolutionize information processing to flexible electronics that seamlessly integrate with biological systems, single-layer graphene shines at the forefront of materials innovation.

The discovery of single-layer graphene reminds us that the most profound technological advances often emerge from the pursuit of fundamental scientific understanding, and that materials as seemingly simple as a single layer of carbon atoms can hold the keys to revolutionary technological capabilities – including entirely new materials, unique properties, and before-never-considered applications.

Thanks for reading!

Appendix:

Notes On The Properties Of Single-Layer Graphene (SLG)

Note: Some property values can vary depending on measurement conditions and sample quality, as factors such as substrate interactions, defects, and environmental conditions can significantly influence measured properties.

Mechanical Properties

• Tensile Strength: Graphene is the strongest material ever tested, with an intrinsic tensile strength of 130 GPa (19,000,000 psi)
• Young’s Modulus: Close to 1 TPa (150,000,000 psi)
• Two-dimensional density: 0.762 mg per square meter
• Theoretical surface area: ~2630 m²/g which is much higher than carbon black (850-900 m²/g) or carbon nanotubes (100-1000 m²/g)
• Spring constant: In the range 1-5 N/m with stiffness of 0.5 TPa

Electrical Properties

• Electron mobility: High electron mobility at room temperature, with values reported in excess of 15000 cm²⋅V⁻¹⋅s⁻¹
• Resistivity: 10⁻⁸ Ω⋅m, lower than the resistivity of silver, which is the lowest known at room temperature
• Zero bandgap: Graphene is a zero-gap semiconductor because its conduction and valence bands meet at the Dirac points
• Ballistic transport: Graphene electrons can traverse micrometer distances without scattering, even at room temperature

Thermal Properties

• Thermal conductivity: Exceptional thermal conductivity of approximately 5300 W⋅m⁻¹⋅K⁻¹ for suspended graphene, compared with pyrolytic graphite of approximately 2000 W⋅m⁻¹⋅K⁻¹ at room temperature
• Substrate effects: When single-layer graphene is supported on an amorphous material, the thermal conductivity is reduced to about 500-600 W⋅m⁻¹⋅K⁻¹ at room temperature due to substrate scattering

Optical Properties

• Light absorption: Absorbs approximately πα ≈ 2.3% of light from visible to infrared wavelengths, where α is the fine-structure constant
• Transparency: Up to 2.3% of white light is absorbed by each layer of graphene with a reflectance of less than 0.1%, making pure single graphene layer highly transparent
• Flexibility: High degree of flexibility along with transparency

Chemical Properties

• Chemical reactivity: Single-layer graphene is a hundred times more chemically reactive than thicker multilayer sheets
• Reaction temperature: The onset temperature of reaction between the basal plane of single-layer graphene and oxygen gas is below 260°C (530 K)
• Burning temperature: Graphene burns at very low temperatures (e.g., 350°C (620 K))

Structural Properties

• Bond length: Molecular bond length of 0.142 nm (1.42 Å)
• Thickness: One atom thick (0.345 nm thick according to one reference)
• Crystal structure: Hexagonal honeycomb lattice of sp²-bonded carbon atoms

Quantum Properties

• Quantum Hall Effect: Shows anomalous quantum Hall effect with conductivity quantization shifted by 1/2 with respect to standard sequence
• Dirac fermions: Electrons behave as massless relativistic particles described by Dirac equation rather than Schrödinger equation

Single-Layer Graphene Property Comparison Tables

Note: Property values can vary depending on measurement conditions, sample quality, defect density, and substrate interactions. The values presented here represent typical or best-case scenarios reported in the literature. Single-layer graphene’s exceptional combination of properties across multiple categories is what makes it truly unique among materials.

Mechanical Properties Comparison

Electrical Properties Comparison

Thermal Properties Comparison

Optical Properties Comparison

Structural Comparison

Glossary Of Key Terms From The Article

2D Material: A crystalline material consisting of a single layer of atoms, with strong in-plane bonding but weak out-of-plane interactions.

Allotrope: Different structural forms of the same element; graphene is a carbon allotrope along with diamond, graphite, and fullerenes.

Anomalous Quantum Hall Effect: The unusual quantum Hall behavior in graphene where conductivity plateaus occur at half-integer values rather than integer values.

Autocorrelation Function: A mathematical tool used to measure how a signal correlates with itself over different time delays, important in thermal conductivity calculations.

Ballistic Transport: The phenomenon where charge carriers (electrons) can travel through a material without scattering from impurities or phonons, allowing for very high conductivity.

Boltzmann Constant (kB): A fundamental physical constant relating temperature to energy, appearing in many thermal property calculations.

Brillouin Zone: The primitive unit cell in reciprocal space, important for describing the electronic band structure of crystalline materials.

Carbon-Carbon Bond Length: The distance between adjacent carbon atoms in graphene, approximately 1.42 Å.

Charge Neutrality Point: The energy level in graphene where the number of electrons equals the number of holes, typically at the Dirac point.

Composite Materials: Materials made from two or more constituent materials with different properties, where graphene can serve as a reinforcing component.

Conductivity Quantization: The phenomenon where electrical conductivity takes on discrete, quantized values in quantum Hall systems.

Defect Formation Energy: The energy required to create a specific type of defect (such as a vacancy) in a crystal lattice.

Dirac Fermions: Particles that obey the Dirac equation, typically massless and relativistic. In graphene, electrons behave as if they are massless Dirac fermions.

Dirac Points: Special points in the Brillouin zone (K and K’ points) where the conduction and valence bands of graphene meet, creating linear energy dispersion.

Exfoliation: The process of separating individual layers from a layered material, used to isolate single-layer graphene from graphite.

Fermi Velocity (vF): The velocity of charge carriers at the Fermi level, particularly important in describing graphene’s linear dispersion.

Fine-Structure Constant (α): A fundamental physical constant approximately equal to 1/137, which appears in graphene’s optical absorption formula.

Fluctuation-Dissipation Theorem: A fundamental principle in statistical mechanics relating spontaneous fluctuations to the response of a system to external perturbations.

Force Convergence Threshold: A computational parameter defining the maximum force allowed on atoms during structural optimization calculations.

Green-Kubo Method: A theoretical approach used in statistical mechanics to calculate transport properties like thermal conductivity from molecular dynamics simulations.

Heat Flux: The rate of thermal energy transfer through a material, measured in watts per square meter.

Hexagonal Lattice: The crystal structure of graphene, where carbon atoms are arranged in a honeycomb pattern with six-fold symmetry.

Hopping Energy: In tight-binding models, the energy associated with an electron moving from one atomic site to a neighboring site.

Interlayer Spacing: The distance between adjacent layers in multilayer graphene or graphite, typically ~3.35 Å.

Intralayer Interstitials: Defects where atoms are inserted between normal lattice positions within a single graphene layer.

Langevin Thermostat: A computational method for controlling temperature in molecular dynamics simulations by adding random and friction forces.

Lattice Constant (a₀): The physical dimension of unit cells in a crystal lattice, for graphene a₀ = 2.47 Å.

Linear Dispersion: The linear relationship between energy and momentum near the Dirac points in graphene’s band structure.

Magic Angle: The specific twist angle (~1.1°) between two graphene layers that creates superconducting properties in twisted bilayer graphene.

Massless Dirac Fermions: Charge carriers in graphene that behave as if they have zero rest mass and follow relativistic physics.

Molecular Dynamics (MD) Simulations: Computational methods that simulate the motion of atoms and molecules over time.

Monolayer: A single layer of atoms or molecules, synonymous with single-layer graphene.

Nudged Elastic Band (NEB) Method: A computational technique used to find minimum energy pathways for chemical reactions and atomic migrations.

Pauli Matrices (σμ): A set of 2×2 complex matrices used in quantum mechanics and essential for describing graphene’s electronic properties.

Phonon: Quantized vibrations of atoms in a crystal lattice that contribute to thermal and mechanical properties.

Phonon-Phonon Scattering: Interactions between phonons that limit thermal conductivity at high temperatures.

Quantum Hall Effect: A quantum mechanical version of the Hall effect where the Hall conductance takes on quantized values.

Quantum Information Processing: The use of quantum mechanical phenomena for computing and information transfer applications.

Reciprocal Space: The space defined by the reciprocal lattice, used to describe electron momentum states and band structures.

Relativistic Behavior: Properties that follow the principles of special relativity, observed in graphene’s charge carriers despite low velocities.

Slater-Koster Formula: A method for calculating hopping integrals in tight-binding models based on atomic orbital overlaps.

sp² Hybridization: The bonding configuration in graphene where each carbon atom forms three covalent bonds in a planar arrangement.

Stone-Wales Defect: A specific type of topological defect in graphene involving the rotation of a carbon-carbon bond.

Sublattice: The division of graphene’s hexagonal lattice into two interpenetrating triangular lattices (A and B sites).

Superconductor: A material that can conduct electricity with zero electrical resistance under certain conditions.

Tensile Strength: The maximum stress a material can withstand while being stretched before breaking.

Tight-Binding Model: A quantum mechanical model used to calculate electronic band structures by considering interactions between nearest-neighbor atoms.

Time-Step: The discrete time interval used in molecular dynamics simulations, typically measured in femtoseconds.

Transparent Conductor: A material that is both electrically conductive and optically transparent, important for display technologies.

Twistronics: The field of study focused on how the electronic properties of layered materials change with the twist angle between layers.

Uniaxial Tension: A type of mechanical stress where force is applied along a single axis.

Vacancy: A point defect where an atom is missing from its normal lattice position.

Valley Degeneracy: The equivalent K and K’ valleys in graphene’s band structure that provide an additional degree of freedom for electrons.

Valleytronic Devices: Electronic devices that exploit the valley degree of freedom in materials like graphene for information processing.

van der Waals Interactions: Weak intermolecular forces that hold separate graphene layers together in multilayer structures.

Van Hove Singularities: Points in the density of electronic states where the density becomes very large due to the geometry of the energy bands.

Velocity Auto-Correlation Function (VACF): A statistical measure that describes the temporal correlations of particle velocities, used to study dynamic properties of materials.

Young’s Modulus: A measure of the stiffness of a material, defined as the ratio of stress to strain in the linear elastic region.

Zero-Gap Semiconductor: A material where the conduction and valence bands meet at a point, resulting in no energy gap between them.