# Graphene **Graphene** is a single layer of [[Carbon|carbon]] atoms in a [[Hexagonal_tiling|hexagonal]] lattice — one sheet of [[Graphite|graphite]], isolated. It is the parent of every other sp² [[Allotropes_of_carbon|carbon allotrope]]: roll it into a [[Carbon_nanotube|carbon nanotube]], wrap it into a [[Fullerene|fullerene]], stack it into graphite. Its interest is not that it is thin but that its electrons are unusual: at the corners of its Brillouin zone the valence and conduction bands meet at a point, and near that point the energy is proportional to momentum rather than to its square, so the carriers behave like massless particles.[^wallace1947][^novoselov2005] In the microsim below the reader gets the whole band structure as a live surface. The sim evaluates the nearest-neighbour tight-binding result `E(k) = ±t·sqrt(1 + 4·cos(sqrt(3)·k_x·a/2)·cos(k_y·a/2) + 4·cos²(k_y·a/2))` over the Brillouin zone on a 128×128 grid, with `a = 0.246 nm`; the control is the hopping energy `t`, or a scrub along a path through the high-symmetry points. The consequence is the [[Dirac_cone|Dirac cones]]: the two sheets touch at the six `K` points, the gap is exactly zero, the dispersion there is linear, and the readout is `v_F = 3·t·a_CC/(2·ħ)`, which for `t = 2.7 eV` is `8.7×10⁵ m/s`, a three-hundredth of the speed of light. On the [[Materials_science]] flagship this page serves Part VII, *Research*, in the section *Graphene*, as the sibling of [[Carbon_nanotube|carbon nanotube]]: the nanotube sim rolls the sheet, this one unrolls it and asks what the roll-up was folding. ## History Graphite's layered structure and its intercalation compounds were long studied; Benjamin Brodie's 1859 work on graphite oxide effectively produced dispersed single layers without recognising them.[^brodie1859] The theory came first: in 1947 Philip Wallace computed graphite's bands by starting from one isolated layer and obtained the linear dispersion at the zone corners, treating it as a step toward the three-dimensional problem rather than as a material.[^wallace1947] The name was proposed by Boehm and co-workers in 1986.[^boehm1986] ### Full isolation and characterization Isolated layers had been seen repeatedly — on metal surfaces, in intercalation residues, as thin flakes — without being measured as a free-standing electronic system. That changed in 2004, when Novoselov, Geim and co-workers cleaved graphite with adhesive tape onto an oxidised [[Silicon|silicon]] wafer, found the flakes by optical contrast and measured an ambipolar [[Electric_field|field effect]] with mobilities above 10⁴ cm²/(V·s) at room temperature.[^novoselov2004] The decisive result came in 2005, when two groups independently found the half-integer quantum Hall effect and a Berry phase of π, the signature of massless carriers.[^novoselov2005][^zhang2005] Geim and Novoselov received the 2010 Nobel Prize in Physics.[^nobel2010] ## Structure Graphene is a two-dimensional [[Crystal_structure|crystal]] with a two-atom basis: the hexagonal lattice is not itself a [[Bravais_lattice|Bravais lattice]] but two interpenetrating triangular sublattices, conventionally A and B. The lattice constant is `a = 0.246 nm` and the carbon–carbon distance `a_CC = a/sqrt(3) = 0.142 nm`. Two atoms in a cell of area `(sqrt(3)/2)·a²` give an areal density of 0.76 mg per square metre. ### Bonding Each atom uses three sp² hybrid orbitals for in-plane σ bonds at 120°, the shortest and strongest [[Covalent_bond|covalent]] bonds carbon forms, and gives its remaining p_z electron to a delocalised π system perpendicular to the sheet. The σ framework carries the mechanical properties and lies far from the Fermi level; the π system carries everything electronic — which is why a one-orbital tight-binding model, the sim's model, works as well as it does. ### Stability A strictly two-dimensional crystal should not exist: the Mermin–Wagner argument shows that long-wavelength thermal fluctuations destroy long-range order in two dimensions at any finite [[Temperature|temperature]].[^mermin1968] Real graphene escapes by not being flat — suspended sheets carry static ripples a few nanometres across, and the third dimension they borrow stabilises the two they have.[^meyer2007] ## Electronic properties The π system is half filled, so the [[Fermi_level|Fermi level]] sits exactly where the bands meet: graphene is neither a metal nor a [[Semiconductor|semiconductor]] but a zero-gap semimetal whose density of states vanishes linearly at the Fermi energy. A gate voltage sweeps the Fermi level through the Dirac point, so one device conducts by electrons above it and holes below it with nearly symmetric characteristics — the ambipolar transport of the 2004 experiment — and [[Electron_mobility|mobility]] can exceed silicon's by more than an order of magnitude in suspended samples.[^novoselov2004] ### Dispersion relation The sim's surface comes from diagonalising a 2×2 matrix whose off-diagonal element sums the three phase factors from an A atom to its B neighbours: `E(k) = ±t·|f(k)|`, which expands to the expression above.[^wallace1947][^likharev-qm3] Three points settle the shape. At the zone centre `Γ` every cosine is 1 and `E = ±3·t`, so the π bands span `6·t`, about 16 eV. At `M` the bracket falls to 1 and `E = ±t`, a saddle point that puts a van Hove singularity into the density of states. At the `K` points the three phase factors cancel exactly and `E = 0` — a touching, not a gap. Expanding about `K` gives `E = ±ħ·v_F·|q|`, linear in the deviation `q`: the massless Dirac form, with `v_F` in place of the speed of light. The sim's control makes this concrete. Raising `t` steepens the cones and raises `v_F` in proportion without ever opening a gap, because the zero at `K` is protected by the equivalence of the two sublattices, not by the size of `t`; break that equivalence — boron nitride underneath, or a bilayer in a perpendicular field — and a gap appears. This is the calculation the [[Carbon_nanotube|nanotube]] page folds: quantising `k` around a circumference either does or does not put an allowed line through `K`. ### Chiral half-integer quantum Hall effect In a strong magnetic field the Landau levels of massless carriers go as `sqrt(N·B)` rather than as `(N + ½)·B`, and one level sits at zero energy, shared between electrons and holes. The Hall conductivity is then quantised as `σ_xy = ±4·(e²/h)·(N + ½)`, the four from spin and valley degeneracy and the half from that zero-energy level.[^novoselov2005][^zhang2005] This is the measurement that proved the Dirac picture, and the level spacing is large enough that it survives to room temperature. ## Interactions and phenomena Graphene is all surface, so everything it touches matters. Its carriers respond to adsorbed molecules strongly enough for single-molecule detection; its [[Van_der_Waals_force|van der Waals]] attraction to a substrate is what makes tape exfoliation work and what holds heterostructures together; and its permittivity, set by the same π electrons, carries a gate-tunable terahertz plasmon. The Casimir force between sheets is likewise carrier-density-dependent — a mechanical force controlled by a voltage. ## Optical properties A single sheet absorbs 2.3 % of normally incident visible light — precisely `π·α` with `α` the fine-structure constant, so the figure is fixed by fundamental constants and not by any material parameter.[^nair2008] Each added layer adds another 2.3 %, which is how flakes are counted optically. The absorption saturates at high intensity, making graphene a broadband saturable absorber for mode-locking [[Laser|lasers]]; and a gate voltage that lifts the Fermi level past half a photon energy blocks interband transitions and makes the sheet transparent, the basis of graphene modulators. ## Excitonic properties In a zero-gap semimetal electron–hole pairs are not bound in the ordinary way; graphene's ultraviolet response is instead dominated by a saddle-point exciton tied to the van Hove singularity at `M`, which pulls the absorption peak below the `2·t` the one-electron picture predicts. Spin is the more useful property: spin–orbit coupling in carbon is weak and the dominant isotope is spinless, so [[Electron|spin]] coherence survives over micrometres — a good spin transport channel, though a poor spin source. ## Magnetic properties Pristine graphene is diamagnetic, like graphite, and has no magnetic order. Moments appear only when the sublattice symmetry is broken: vacancies, adatoms and zigzag edges each create localised zero-energy states that can carry spin, and their coupling depends on which sublattice they sit on. A magnetic substrate can instead imprint an exchange field by proximity without adding defects, the cleaner route to spin-polarised graphene. ## Mechanical properties Nanoindentation of suspended membranes gives a [[Young's_modulus|Young's modulus]] near 1 TPa and an intrinsic breaking strength of about 130 GPa, the highest measured for any material and close to the theoretical limit for the C–C bond.[^lee2008] That strength belongs to a defect-free micrometre-scale membrane. Real sheets are polycrystalline, and a [[Grain_boundary|grain boundary]] is a line of pentagon–heptagon pairs whose strength depends on the misorientation, so a large grown film is limited by its boundaries rather than by its bonds. Graphene is also elastic to failure with no plasticity, so toughness rather than strength limits what a graphene composite delivers.[^lee2008] ## Other properties Thermal transport is exceptional: suspended single-layer graphene has a measured in-plane [[Thermal_conductivity_and_resistivity|thermal conductivity]] of several thousand watts per metre-kelvin at room temperature, above [[Diamond|diamond]]'s, carried by [[Phonon|phonons]] on the stiff σ framework.[^balandin2008] A substrate cuts this sharply by damping the flexural modes that carry much of the heat. The pristine basal plane is chemically inert, having no dangling bonds, and reactivity concentrates at edges and defects; curvature raises it, which is why a nanotube is more reactive than a flat sheet. The support substrate matters more than any of this: hexagonal boron nitride, atomically flat and free of dangling bonds, raises mobility by an order of magnitude over [[Silicon_dioxide|silicon dioxide]]. ## Graphene layers and structural variants A monolayer is a Dirac semimetal; two layers are not. Bernal-stacked bilayer graphene has parabolic bands touching at zero energy — massive chiral carriers — and a perpendicular electric field breaks the equivalence of the layers, opening a gap of up to a few hundred millielectronvolts: the gate-tunable [[Band_gap|band gap]] a monolayer cannot provide. Randomly rotated turbostratic stacks are decoupled and behave like many independent monolayers. Rotating two layers by about 1.1° instead flattens the moiré bands and produces correlated insulating states and [[Superconductivity|superconductivity]].[^cao2018] ## Nanostructured graphene forms Confining graphene in one or both directions restores a gap, which is the standard answer to the zero-gap problem. ### Graphene nanoribbons A ribbon of width `W` quantises the transverse momentum and opens a gap scaling roughly as `1/W`, so a gap near 1 eV — enough for a room-temperature switch — needs a ribbon a few nanometres wide with atomically precise edges. Edge geometry matters as much as width: armchair ribbons are metallic or semiconducting according to width modulo three, exactly as nanotubes are, while zigzag ribbons carry localised zero-energy edge states. Bottom-up synthesis from designed molecular precursors reaches edges lithography cannot. [[Quantum_dot|Quantum dots]] confine in both directions for a size-tunable gap. ## Modified and functionalized graphene Graphene oxide is the industrially important derivative: graphite oxidised by the Brodie, Staudenmaier or Hummers route becomes a hydrophilic solid bearing epoxide, hydroxyl and carboxyl groups that exfoliates in water into single layers.[^brodie1859][^hummers1958] It is an insulator, and chemical or thermal reduction restores conduction only partly because the sp² network never fully heals — reduced graphene oxide is a different material and should be judged as one. Covalent basal-plane chemistry (hydrogenation to graphane, fluorination) opens a large gap at the cost of the Dirac physics; π-stacking leaves the bands intact. ## Advanced graphene structures Graphene can be assembled upward. Wet-spinning graphene-oxide liquid crystals through a coagulation bath and reducing the result gives continuous fibres, competitive in thermal and electrical conductivity if not in strength. Foams grown on nickel templates, pillared architectures in which nanotubes space the sheets apart, and metal- or [[Polymer|polymer]]-infiltrated laminates all attack one problem: a sheet is useful in bulk only if the sheets are connected. Load transfer between sheets, not the strength of any sheet, limits every one of them. ## Specialized graphene configurations Graphene [[Aerogel|aerogels]], made by freeze-drying or critical-point drying a graphene-oxide hydrogel, are among the lightest solids made and serve as absorbents and [[Supercapacitor|supercapacitor]] electrodes. Nanocoils are helical ribbons with a geometric inductance. Crumpled graphene resists the restacking that is the practical failure mode of every graphene powder, keeping more surface area accessible in an electrode. ## Mechanical synthesis Top-down routes separate graphite's layers against a van der Waals binding energy of a few tens of millielectronvolts per atom. Micro-mechanical cleavage with adhesive tape, the method of the 2004 experiment, still gives the best crystals and the smallest yields.[^novoselov2004] Liquid-phase exfoliation — sonicating or shearing graphite in a solvent whose surface energy matches graphite's — is the scalable version and produces most commercial graphene, as a dispersion of few-layer flakes about a micrometre across. Ball milling and unzipping nanotubes lengthwise give ribbons and small flakes. ## Chemical synthesis The chemical routes go through graphite oxide. Oxidation inserts oxygen functionalities that swell the interlayer spacing, exfoliation in water is then trivial, and reduction by hydrazine, ascorbic acid or heating restores part of the conjugation.[^hummers1958] Variants replace the oxidant with molten salts or an electrochemical cell, where intercalating ions drive expansion directly and the material never becomes fully oxidised. Bottom-up organic synthesis from polycyclic aromatic precursors builds defined nanographenes rather than large sheets. ## Vapor deposition and growth techniques Chemical vapour deposition on copper is the route to large continuous films. Copper's negligible carbon solubility makes growth self-limiting at one layer: methane decomposes on the surface, carbon [[Nucleation|nucleates]] into islands that expand and merge, and the reaction stops once the surface is covered. The film is polycrystalline and must be transferred off the [[Catalysis|catalyst]] onto a target substrate — the step that causes most of the contamination and tearing seen in practice. Thermal decomposition of silicon carbide inverts the problem: heating a SiC wafer sublimes silicon and leaves graphitised carbon on an insulator, with no transfer at all. ## Simulation Graphene is unusually well served by simple models. The tight-binding Hamiltonian the sim evaluates is a two-band model with one parameter, and it reproduces the π structure well enough for transport; density-functional theory supplies the σ bands, elastic constants and adsorption energies; molecular dynamics with bond-order potentials handles fracture and ripples. The underlying machinery is the standard [[Particle_in_a_one-dimensional_lattice|periodic-potential]] apparatus — Bloch's theorem, quasimomentum, band width, effective mass — which the Portal Books develop in one dimension before it is extended to a two-atom basis.[^likharev-qm3] ## Graphene analogs The construction generalises. [[Silicene|Silicene]], germanene and stanene are the group-14 analogues, buckled rather than flat and with stronger spin–orbit coupling; hexagonal boron nitride is the same lattice with two different atoms, which breaks the sublattice symmetry and opens a 6 eV gap, making it graphene's natural insulating substrate; the transition-metal dichalcogenides and MXenes extend the family to semiconductors and metals. Stacking these into heterostructures is now a design method in itself. ## Applications The realised applications are bulk ones. Graphene and reduced graphene oxide are conductive additives in [[Lithium-ion_battery|battery]] and [[Supercapacitor|supercapacitor]] electrodes, reinforcements and barrier fillers in [[Composite_material|composites]] and coatings, and additives in concrete and inks; a little flake in a [[Percolation|percolating]] network changes a polymer's conductivity by orders of magnitude. Transparent conductive films, flexible electrodes and [[Sensor|sensors]] exploiting an all-surface conductor are in small-scale production. Digital electronics remains blocked by the missing gap, so effort has moved to radio-frequency devices, where transconductance matters more than the on/off ratio, and to [[Transistor|transistors]] built from bilayers and ribbons. ## Toxicity Graphene materials are not one substance for toxicological purposes, and the literature is coherent only when lateral size, layer number, oxidation state, functionalisation and residual catalyst are reported. Large rigid few-layer flakes raise the fibre-like concerns that apply to [[Carbon_nanotube|nanotubes]]; small, well-dispersed, heavily oxidised material behaves differently and clears more readily. Occupational practice treats graphene powders as respirable [[Nanomaterials|nanomaterials]] and controls dust, which is necessary in any case because the powders are hard to handle. ## See also - [[Dirac_cone]] - [[Silicene]] - [[Graphite]] - [[Hexagonal_tiling]] - [[Carbon_nanotube]], the same sheet rolled - [[Allotropes_of_carbon]] - [[Nanomaterials]] - [[Electronic_band_structure]] ## References [^wallace1947]: Wallace, P. R. (1947). "The Band Theory of Graphite." *Physical Review* 71 (9): 622–634. https://doi.org/10.1103/PhysRev.71.622 The nearest-neighbour tight-binding dispersion the sim evaluates, and the linear behaviour at the zone corners. [^brodie1859]: Brodie, B. C. (1859). "On the Atomic Weight of Graphite." *Philosophical Transactions of the Royal Society of London* 149 (page to pin). [^boehm1986]: Boehm, H. P.; Setton, R.; Stumpp, E. (1986). "Nomenclature and terminology of graphite intercalation compounds." *Carbon* 24 (page to pin). The proposal of the name *graphene* for a single layer of the graphite structure. [^mermin1968]: Mermin, N. D. (1968). "Crystalline Order in Two Dimensions." *Physical Review* 176 (1): 250–254. https://doi.org/10.1103/PhysRev.176.250 [^meyer2007]: Meyer, J. C.; Geim, A. K.; Katsnelson, M. I.; Novoselov, K. S.; Booth, T. J.; Roth, S. (2007). "The structure of suspended graphene sheets." *Nature* 446 (page to pin). The static ripples that reconcile a two-dimensional crystal with the Mermin–Wagner argument. [^novoselov2004]: Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. (2004). "Electric Field Effect in Atomically Thin Carbon Films." *Science* 306 (5696): 666–669. https://doi.org/10.1126/science.1102896 [^novoselov2005]: Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. (2005). "Two-dimensional gas of massless Dirac fermions in graphene." *Nature* 438 (7065): 197–200. https://doi.org/10.1038/nature04233 [^zhang2005]: Zhang, Y.; Tan, Y.-W.; Stormer, H. L.; Kim, P. (2005). "Experimental observation of the quantum Hall effect and Berry's phase in graphene." *Nature* 438 (7065): 201–204. https://doi.org/10.1038/nature04235 [^lee2008]: Lee, C.; Wei, X.; Kysar, J. W.; Hone, J. (2008). "Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene." *Science* 321 (5887): 385–388. https://doi.org/10.1126/science.1157996 [^nair2008]: Nair, R. R.; Blake, P.; Grigorenko, A. N.; Novoselov, K. S.; Booth, T. J.; Stauber, T.; Peres, N. M. R.; Geim, A. K. (2008). "Fine Structure Constant Defines Visual Transparency of Graphene." *Science* 320 (5881): 1308. https://doi.org/10.1126/science.1156965 [^balandin2008]: Balandin, A. A.; Ghosh, S.; Bao, W.; Calizo, I.; Teweldebrhan, D.; Miao, F.; Lau, C. N. (2008). "Superior Thermal Conductivity of Single-Layer Graphene." *Nano Letters* 8 (3): 902–907. https://doi.org/10.1021/nl0731872 [^hummers1958]: Hummers, W. S.; Offeman, R. E. (1958). "Preparation of Graphitic Oxide." *Journal of the American Chemical Society* 80 (page to pin). [^cao2018]: Cao, Y.; Fatemi, V.; Fang, S.; Watanabe, K.; Taniguchi, T.; Kaxiras, E.; Jarillo-Herrero, P. (2018). "Unconventional superconductivity in magic-angle graphene superlattices," and the companion paper on correlated insulator behaviour. *Nature* 556 (page to pin). [^nobel2010]: Royal Swedish Academy of Sciences (2010). *The Nobel Prize in Physics 2010: Andre Geim and Konstantin Novoselov, "for groundbreaking experiments regarding the two-dimensional material graphene."* Press release and scientific background (page to pin). [^likharev-qm3]: Likharev, K. *Essential Graduate Physics, Part QM: Quantum Mechanics* (2013), Ch. 3, pp. 107–170: Bloch's theorem `ψ(x + a) = ψ(x)·exp(i·q·a)` and the quasimomentum, the tight-binding band `E = E_n + 2·ħ·η·cos(q·a)` of width `4·ħ|η|` with its validity bound `ħ|η_n| ≪ E_n`, the gaps of the weak-potential limit, and the effective mass `1/m_ef = (1/ħ²)·d²E/dq²` — the one-dimensional periodic-lattice machinery that the two-atom basis extends into the sim's two-dimensional surface (page to pin). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^geom-note]: The structural and band numbers quoted here are computed in this article from the row's own constants and are derived, not measured: `a_CC = a/sqrt(3) = 0.142 nm` from `a = 0.246 nm`; the areal density 0.76 mg/m² from two atoms per cell of area `(sqrt(3)/2)·a²` and carbon's atomic mass; `E = ±3·t = ±8.1 eV` at `Γ` and `E = ±t` at `M` from the sim's own expression; `v_F = 3·t·a_CC/(2·ħ) = 8.7×10⁵ m/s` and the ratio `c/v_F ≈ 343` for `t = 2.7 eV`; and the 2.3 % absorption as `π·α` with `α = 1/137.036`, which is the measured result of Nair et al. rather than a fit. ## External links - The Wikipedia pair's *External links* section lists the current graphene standards bodies, producer registries and band-structure applets; none is reproduced here until its URL has been checked. - Likharev, *Part QM*, is on the Open Textbook Library (link in the references above). <!-- MATTERSIM:BEGIN g24 — Matter & Energy Cluster microsim (framework build, specs/sims/Graphene.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework), pending deploy:** *Graphene* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Graphene.html` is live. <!-- pending: <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Graphene.html" data-title="Graphene"></div> --> <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Graphene) : [Wikitube](https://en.wikitube.io/wiki/Graphene) · pinned revision [1373529087](https://en.wikipedia.org/w/index.php?oldid=1373529087) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M35 · sim pending (matter/Graphene).*