# Carbon nanotube A **carbon nanotube** is a tube of [[Carbon|carbon]] whose wall is a single sheet of atoms: a strip of the hexagonal [[Graphene|graphene]] lattice, wrapped so that its two edges join without a seam. It is one of the [[Allotropes_of_carbon|allotropes of carbon]], sitting between the closed cages of the [[Fullerene|fullerenes]] and the flat sheets of [[Graphite|graphite]], and it is a genuinely one-dimensional object — a nanometre or so across and, in the best samples, centimetres long.[^saito1998] Almost everything interesting about it follows from the one arbitrary choice made when the sheet is rolled. In the microsim below that choice is the control. The reader sets the two integers `(n, m)` of the chiral vector, the sheet rolls along it in three dimensions, and the tube rebuilds. The readouts are the diameter `d = a·sqrt(n² + n·m + m²)/π` with `a = 0.246 nm`, the chiral angle `θ = atan(sqrt(3)·m/(2·n + m))`, and a verdict the reader can watch flip: a tube is metallic if and only if `(n − m) mod 3 = 0`, and semiconducting otherwise.[^hamada1992][^mintmire1992] Presets place the two achiral families — armchair `(n, n)` at `θ = 30°`, always metallic, and zigzag `(n, 0)` at `θ = 0°` — at the ends of the range, with every chiral tube between them. On the [[Materials_science]] flagship this page serves Part VII, *Research*, in the section *Nanomaterials*, where it is the parent of the [[Graphene|graphene]] sibling: unroll the tube and the same tight-binding calculation gives the flat sheet's [[Dirac_cone|Dirac cones]], and the metallic rule is simply the question of whether the rolled-up tube's allowed lines in reciprocal space happen to pass through one. ## History Hollow carbon filaments were seen long before they were understood: Radushkevich and Lukyanovich published electron micrographs of tubular carbon in 1952, and Oberlin, Endo and Koyama reported vapour-grown fibres with hollow cores in 1976.[^radushkevich1952][^oberlin1976] The subject began in earnest in 1991, when Sumio Iijima described "helical microtubules of graphitic carbon" in the cathode deposit of an arc-discharge apparatus being used to make fullerenes, and named the helicity of the lattice as the defining variable.[^iijima1991] Two years later Iijima and Ichihashi, and independently Bethune and co-workers, reported single-walled tubes about a nanometre across, grown with metal catalysts.[^iijima1993][^bethune1993] Theory moved faster than synthesis. Within months of the 1991 paper, three groups folded graphene's bands onto the allowed states of a cylinder and reached the same startling conclusion: whether a tube conducts like a metal or a [[Semiconductor|semiconductor]] is fixed by arithmetic on `(n, m)` alone, with no change of chemistry whatever.[^hamada1992][^mintmire1992][^saito1998] It went unverified for six years, until scanning tunnelling spectroscopy on structurally resolved tubes confirmed it in 1998.[^wilder1998] ## Structure of SWCNTs The construction the sim animates is the standard one. Take the graphene lattice with primitive vectors `a₁` and `a₂` of length `a = 0.246 nm` at 60°, choose a chiral vector `C_h = n·a₁ + m·a₂` joining two equivalent atoms, and roll the sheet until the ends of `C_h` coincide. `C_h` is then the circumference, so `d = |C_h|/π = a·sqrt(n² + n·m + m²)/π`, with the tube axis perpendicular to `C_h`.[^saito1998] The arithmetic is worth doing once. For the armchair `(10,10)` tube, `sqrt(300) = 17.32` and `d = 0.246 × 17.32/π = 1.36 nm`; for the zigzag `(10,0)`, `d = 0.78 nm`; for the chiral `(6,5)`, `sqrt(91) = 9.54` gives `d = 0.75 nm` and `θ = atan(5·sqrt(3)/17) = 27.0°`. Only integer pairs are allowed, so diameters form a discrete set, and two tubes of nearly equal diameter can have completely different [[Electronic_band_structure|band structures]] — which is why the sim's verdict line matters more than its geometry readout. ### Basic details Every atom in the wall is three-fold coordinated and sp²-bonded as in graphene, with a [[Covalent_bond|covalent]] C–C bond length of about 0.142 nm; `a = sqrt(3) × 0.142 nm` is that bond length scaled to the lattice constant. Curvature spoils the planar sp² geometry slightly, mixing in sp³ character, which makes a nanotube more reactive than flat graphene and opens a small gap in tubes the simple rule calls metallic.[^saito1998] A closed tube is capped by half a fullerene at each end, needing exactly six pentagons per cap; oxidation opens the ends, the starting point for most chemistry. ## Types Tubes are classified by `(n, m)`, and the classification is exhaustive: three families cover every possibility, and the sim's two sliders move continuously through all of them. ### Chirality and mirror symmetry Armchair tubes have `m = n`, and the atoms around the circumference trace the armchair edge of the hexagonal lattice; `θ = atan(sqrt(3)·n/(3·n)) = 30°`, the maximum. Zigzag tubes have `m = 0` and `θ = 0°`. Everything else is chiral, with `0 < θ < 30°`, and the lattice winds around the tube like a helix. A chiral tube is not superimposable on its mirror image: `(n, m)` and `(m, n)` are enantiomers, with identical diameters and band structures but opposite optical activity, while armchair and zigzag tubes are their own mirror images. This is [[Chirality_(chemistry)|chirality]] in the strict chemical sense, and since synthesis has no reason to prefer one hand, as-grown material is racemic. ### Circumference and diameter The circumference `|C_h| = a·sqrt(n² + n·m + m²)` is the only length the roll-up introduces, and it controls more than size. The [[Band_gap|band gap]] of a semiconducting tube scales as the inverse diameter — roughly `0.77 eV·nm / d` in the simple tight-binding picture, so about 1.0 eV for the `(10,0)` tube and 0.57 eV for a 1.36 nm one.[^saito1998] Diameter also fixes the density. A graphene sheet weighs 0.76 mg per square metre, so a `(10,10)` tube weighs `π·d × 0.76 mg/m² = 3.2 ng` per metre of length; packed into a bundle at the [[Van_der_Waals_force|van der Waals]] spacing of 0.34 nm, that comes to about 1.3 g/cm³, roughly half the density of graphite. ## Physical limits The family has hard ends, and all three are set by geometry rather than by chemistry. ### Narrowest examples Curvature costs strain energy that rises as `1/d²`, so very narrow tubes are expensive to make and unstable once made. The observed floor is about 0.4 nm — the `(5,0)`, `(3,3)` and `(4,2)` tubes — reached only as the innermost shell of a multi-walled tube or grown inside a zeolite channel that templates the diameter.[^saito1998] ### Length There is no upper limit in principle: a nanotube is a one-dimensional crystal and grows until its catalyst particle dies. Centimetre-long individual tubes have been grown by chemical vapour deposition, giving aspect ratios above 10⁷ — which is what makes nanotubes so effective at forming a conducting [[Percolation|percolation]] network at very low loadings in a [[Composite_material|composite]]. ### Density The useful density is the bundle's, not the tube's, because tubes stick together. From first principles: a graphene wall has an areal density of 0.76 mg/m², so a tube of diameter `d` has linear density `π·d × 0.76 mg/m²`, and hexagonally packed tubes at centre-to-centre spacing `d + 0.34 nm` each occupy an area `(sqrt(3)/2)·(d + 0.34)²`. For `d = 1.36 nm` that gives 1.3 g/cm³, against 2.27 for graphite and 3.5 for [[Diamond|diamond]] — light for a carbon material, which is the whole basis of the specific-strength argument. ## Variants The single-walled tube is the idealisation; most nanotubes in the world are not one. ### Multi-walled A multi-walled nanotube is a set of concentric tubes separated by about 0.34 nm, the [[Graphite|graphite]] interlayer spacing, held by van der Waals forces. Adjacent shells generally have incommensurate chiralities, so they couple only weakly and the inner shells slide and telescope almost without friction. The double-walled tube is the limiting case, valued because the outer wall can be functionalised while the inner one keeps clean electronic properties. ### Junctions and crosslinking A pentagon–heptagon pair in the wall changes the chirality on one side of it, so a single topological defect can join a metallic tube to a semiconducting one and make a rectifying junction out of pure carbon; larger defect arrangements give Y- and T-junctions. Deliberate crosslinking between tubes, by irradiation or chemistry, converts a weakly bound bundle — which fails by tubes sliding past one another — into something that must break bonds to fail, and that is the central problem in turning nanotube strength into fibre strength. ### Other morphologies The same wrapping idea generates a family: cup-stacked and bamboo tubes, nanohorns, nanobuds with [[Fullerene|fullerenes]] attached to the sidewall, nanotorus rings, and carbon [[Nanoparticle|nanoparticles]] built from curved graphitic shells. Peapods — fullerenes inside a tube — show that the interior is a usable container. ## Properties The properties divide cleanly: the mechanical ones follow from the strength of the sp² bond, and the electronic ones from the roll-up arithmetic the sim computes. ### Mechanical In-plane sp² bonding is the stiffest arrangement carbon has, and a nanotube inherits it along its axis. Measured [[Young's_modulus|Young's moduli]] are near 1 TPa, five times steel's, and tensile tests on individual multi-walled tubes gave breaking strengths of tens of gigapascals, an [[Ultimate_tensile_strength|ultimate strength]] far above any bulk [[Strength_of_materials|structural material]].[^yu2000] They fail by the outermost shell breaking and the inner ones pulling out, a "sword-in-sheath" failure that limits how much strength a real bundle delivers.[^yu2000] ### Electrical The metallic rule is the sim's punchline. All armchair tubes satisfy `(n − m) mod 3 = 0` and so are always metallic; among the rest one in three is metallic, so an uncontrolled synthesis yields about a third metallic material.[^hamada1992][^mintmire1992] A metallic tube has two conducting bands, each spin-degenerate, so its ideal conductance is `4·e²/h` and even a perfect tube with perfect contacts has a resistance of about 6.5 kΩ. Transport along a clean tube is ballistic over micrometres and [[Electron_mobility|mobilities]] in semiconducting tubes are far above [[Silicon|silicon]]'s, the basis of every nanotube [[Transistor|transistor]] proposal. ### Optical A semiconducting tube absorbs and emits at energies set by the van Hove singularities of its one-dimensional bands, and those energies depend on `(n, m)`. Plotting emission against excitation wavelength gives a map in which each chirality is its own spot — how a sample's chirality distribution is measured without imaging a tube.[^dresselhaus2001] ### Thermal Heat travels along the axis by [[Phonon|phonons]] on the same stiff bonds, and axial [[Thermal_conductivity_and_resistivity|thermal conductivity]] is among the highest measured for any material, comparable with [[Synthetic_diamond|diamond]]'s, while conduction across a bundle is worse by a factor of hundreds.[^dresselhaus2001] Nanotubes are therefore strongly anisotropic thermal conductors, useful at an interface only when aligned across it. ## Synthesis Three routes dominate. Arc discharge between graphite electrodes, the method of the original discovery, gives highly crystalline tubes mixed with soot and needs a metal catalyst in the anode for single-walled material.[^iijima1991][^iijima1993] [[Laser|Laser]] ablation of a catalyst-loaded graphite target in a hot furnace gives high-quality single-walled tubes at low yield. Chemical vapour deposition — decomposing a hydrocarbon or carbon monoxide over iron, cobalt or nickel [[Catalysis|catalyst]] particles at 600–1,000 °C — is the only route that scales and produces almost all commercial material; patterning the catalyst controls position and alignment, and catalyst particle size sets tube diameter, the one structural handle synthesis currently offers. ## Purification As-grown material contains amorphous carbon, graphitic shells and catalyst metal. Oxidation in air or acid removes the disordered carbon preferentially because it is more reactive, and acid leaching removes exposed metal, but both attack the tubes and open their ends. Density-gradient ultracentrifugation, selective polymer or DNA wrapping and aqueous two-phase extraction sort by diameter, by metal-versus-semiconductor character, and at best by chirality. ### Advantages of monochiral CNTs Monochiral material matters because the two-thirds/one-third mixture is fatal to electronics: metallic tubes short out a channel built from the semiconducting ones, so a device-grade film needs better than 99.9 % semiconducting purity. A monochiral sample also has one sharp optical transition instead of a smear, and one band gap instead of a distribution. ## Functionalization Chemistry on a nanotube is either covalent or not, and the choice is a trade. Covalent attachment — oxidation to carboxyl groups at the ends and defect sites, then amidation — is robust but converts sp² carbon to sp³ and destroys the electronic structure locally. Non-covalent functionalisation wraps the tube in surfactants or aromatic molecules binding by π-stacking, leaving the bands intact but reversibly. Both solve the same problem: pristine nanotubes are insoluble in everything and are held in bundles by van der Waals attraction too strong for ordinary dispersion. ## Modeling The standard model is zone folding, the calculation the [[Graphene|graphene]] sibling page performs. Compute graphene's π-band dispersion by tight binding, then impose the periodic boundary condition around the circumference, `C_h·k = 2·π·q`, which allows only a set of parallel lines in [[Reciprocal_lattice|reciprocal space]].[^wallace1947][^saito1998] Graphene's bands touch only at the `K` points, so the tube is metallic if an allowed line passes through a `K` point and semiconducting if none does — and the geometry makes that condition precisely `(n − m) mod 3 = 0`.[^hamada1992][^saito1998] Zone folding ignores curvature; including it opens gaps of tens of millielectronvolts in tubes the rule calls metallic, except in armchair tubes, which symmetry protects. Beyond tight binding this is the standard [[Particle_in_a_one-dimensional_lattice|periodic-lattice]] problem whose Bloch machinery the Portal Books develop.[^likharev-qm3] ## Metrology Raman [[Spectroscopy|spectroscopy]] is the workhorse, because a nanotube has a radial breathing mode in which the whole cylinder expands and contracts at a frequency inversely proportional to the diameter — a direct optical measurement of `d`. The ratio of the disorder-induced D band to the graphitic G band grades sample quality. Electron microscopy resolves and counts individual walls; [[Scanning_tunneling_microscope|scanning tunnelling microscopy]] with spectroscopy, the technique that confirmed the metallic rule, gives the lattice and the density of states on the same tube.[^wilder1998] Thermogravimetry gives residual catalyst as the mass left after the carbon burns away. ## Safety and health Nanotubes are biopersistent fibres of high aspect ratio, and long straight ones have the dimensions that make a fibre hazardous in the lung — the geometric argument that governs mineral fibres. Long multi-walled nanotubes introduced into the abdominal cavity of mice produced inflammation and granulomas of the type associated with pathogenic fibres; short or tangled tubes did not.[^poland2008] The controlling variables are therefore length, rigidity, biopersistence and residual metal catalyst, not "nanotube" as a category, and functionalisation that shortens or disperses the material reduces the response. Occupational practice follows this: the US National Institute for Occupational Safety and Health recommends an exposure limit of 1 µg/m³ of elemental carbon as an eight-hour time-weighted average for carbon nanotubes and nanofibres.[^niosh2013] ## Applications Most commercial tonnage goes into things needing conductivity rather than strength: nanotubes are the standard conductive additive in [[Lithium-ion_battery|lithium-ion battery]] electrodes, where a fraction of a percent by mass builds a percolating network without displacing active material, and they appear in dissipative plastics and [[Supercapacitor|supercapacitor]] electrodes. Structural use is modest — sporting goods, and [[Carbon_fiber_reinforced_polymer|carbon-fibre composites]] toughened between the plies — because fibre strength is limited by tubes sliding past one another, not by the tubes. Electronics remains the long project. A complete microprocessor built from nanotube transistors was demonstrated in 2019, on the strength of purification good enough to suppress metallic tubes. Transparent conductive films, field-emission sources, thermal interface materials, [[Sensor|sensors]] that turn a binding event into a change in channel conductance, and aligned-tube membranes are all past demonstration and short of volume. Almost all depend on one unsolved problem: making tubes of a single chirality at scale — the problem the sim's two integers pose. ## See also - [[Nanomaterials]] - [[Fullerene]] - [[Buckminsterfullerene]] - [[Nanotechnology]] - [[Carbon]] - [[Graphene]], the unrolled sheet whose bands the roll-up folds - [[Allotropes_of_carbon]] - [[Hexagonal_tiling]], the lattice being wrapped - [[Dirac_cone]] ## References [^iijima1991]: Iijima, S. (1991). "Helical microtubules of graphitic carbon." *Nature* 354 (6348): 56–58. https://doi.org/10.1038/354056a0 [^iijima1993]: Iijima, S.; Ichihashi, T. (1993). "Single-shell carbon nanotubes of 1-nm diameter." *Nature* 363 (6430): 603–605. https://doi.org/10.1038/363603a0 [^bethune1993]: Bethune, D. S.; Kiang, C. H.; de Vries, M. S.; Gorman, G.; Savoy, R.; Vazquez, J.; Beyers, R. (1993). "Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls." *Nature* 363 (6430): 605–607. https://doi.org/10.1038/363605a0 [^radushkevich1952]: Radushkevich, L. V.; Lukyanovich, V. M. (1952). "O strukture ugleroda, obrazujucegosja pri termiceskom razlozenii okisi ugleroda na zeleznom kontakte." *Zurnal Fizicheskoi Khimii* 26 (page to pin). [^oberlin1976]: Oberlin, A.; Endo, M.; Koyama, T. (1976). "Filamentous growth of carbon through benzene decomposition." *Journal of Crystal Growth* 32 (page to pin). [^hamada1992]: Hamada, N.; Sawada, S.; Oshiyama, A. (1992). "New one-dimensional conductors: Graphitic microtubules." *Physical Review Letters* 68 (10): 1579–1581. https://doi.org/10.1103/PhysRevLett.68.1579 [^mintmire1992]: Mintmire, J. W.; Dunlap, B. I.; White, C. T. (1992). "Are fullerene tubules metallic?" *Physical Review Letters* 68 (5): 631–634. https://doi.org/10.1103/PhysRevLett.68.631 [^wilder1998]: Wilder, J. W. G.; Venema, L. C.; Rinzler, A. G.; Smalley, R. E.; Dekker, C. (1998). "Electronic structure of atomically resolved carbon nanotubes." *Nature* 391 (page to pin); and Odom, T. W.; Huang, J.-L.; Kim, P.; Lieber, C. M. (1998). "Atomic structure and electronic properties of single-walled carbon nanotubes." *Nature* 391 (page to pin). [^yu2000]: Yu, M.-F.; Lourie, O.; Dyer, M. J.; Moloni, K.; Kelly, T. F.; Ruoff, R. S. (2000). "Strength and Breaking Mechanism of Multiwalled Carbon Nanotubes Under Tensile Load." *Science* 287 (5453): 637–640. https://doi.org/10.1126/science.287.5453.637 [^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 π-band dispersion that zone folding folds. [^poland2008]: Poland, C. A.; Duffin, R.; Kinloch, I.; Maynard, A.; Wallace, W. A. H.; Seaton, A.; Stone, V.; Brown, S.; MacNee, W.; Donaldson, K. (2008). "Carbon nanotubes introduced into the abdominal cavity of mice show asbestos-like pathogenicity in a pilot study." *Nature Nanotechnology* 3 (page to pin). [^niosh2013]: National Institute for Occupational Safety and Health (2013). *Current Intelligence Bulletin 65: Occupational Exposure to Carbon Nanotubes and Nanofibers*. DHHS (NIOSH) Publication No. 2013-145 (page to pin). [^saito1998]: Saito, R.; Dresselhaus, G.; Dresselhaus, M. S. *Physical Properties of Carbon Nanotubes* (1998), Imperial College Press, Ch. 3 (the chiral vector, `d` and `θ`, the classification into armchair, zigzag and chiral) and Ch. 4 (zone folding, the `(n − m) mod 3` rule, curvature gaps, the inverse-diameter band gap) (page to pin). No Portal Book covers nanotubes, so this is the reference text for the structural and electronic statements above. [^dresselhaus2001]: Dresselhaus, M. S.; Dresselhaus, G.; Avouris, P. (eds.) *Carbon Nanotubes: Synthesis, Structure, Properties, and Applications* (2001), Springer, Topics in Applied Physics 80 — chapters on thermal transport and on optical spectroscopy, including the van Hove structure behind photoluminescence excitation maps (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)`, the quasimomentum, the tight-binding band `E = E_n + 2·ħ·η·cos(q·a)` of width `4·ħ|η|` and its validity bound, and the effective mass `1/m_ef = (1/ħ²)·d²E/dq²` — the periodic-lattice machinery zone folding applies to a rolled sheet (page to pin). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^geom-note]: The diameters, chiral angles, band gaps, linear densities and bundle density quoted in this article are computed here from the row's own constants (`a = 0.246 nm`, the 0.142 nm C–C bond, the 0.34 nm van der Waals spacing and carbon's atomic mass) and are derived values, not measurements: `d(10,10) = 1.36 nm`, `d(10,0) = 0.78 nm`, `d(6,5) = 0.75 nm` with `θ = 27.0°`; graphene areal density 0.76 mg/m²; bundle density 1.3 g/cm³ at `d = 1.36 nm`; ideal conductance `4·e²/h`, i.e. 6.5 kΩ. ## External links - The Wikipedia pair's *External links* section lists the current nanotube databases, chirality calculators and standards pages; 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/Carbon_nanotube.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework), pending deploy:** *Carbon nanotube* will play here once `https://wikitube-3d-microsims.netlify.app/matter/Carbon_nanotube.html` is live. <!-- pending: <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Carbon_nanotube.html" data-title="Carbon nanotube"></div> --> <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Carbon_nanotube) : [Wikitube](https://en.wikitube.io/wiki/Carbon_nanotube) · pinned revision [1373012818](https://en.wikipedia.org/w/index.php?oldid=1373012818) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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