Carbon Nanotubes: Metallic or Semiconducting?

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By Charles Xie ✉

A carbon nanotube is a sheet of graphene rolled into a cylinder about a nanometer across. Iijima reported the first ones in 1991. Within a year, Hamada, Sawada and Oshiyama, and independently Saito, Fujita, Dresselhaus and Dresselhaus, predicted from band theory that the way the sheet is rolled decides whether the nanotube is metallic or semiconducting. The roll is described by two integers (n, m) that say how many hexagons the circumference spans along the two directions of the honeycomb lattice. (n, 0) nanotubes are called zigzag and (n, n) nanotubes armchair, after the pattern of their rims. The prediction was that a nanotube is metallic when n − m is a multiple of three and semiconducting otherwise, so that one nanotube in three, taken at random, is metallic. In 1998 two groups, Wildöer and coworkers in Delft and Odom and coworkers at Harvard, confirmed the rule on individual nanotubes with a scanning tunneling microscope, and found that the gap of a semiconducting nanotube is inversely proportional to its diameter, as predicted.

The usual textbook treatment is a picture of the graphene Brillouin zone with lines drawn across it. In AIMS a student can test the rule on nanotubes they choose. They name a nanotube by its two integers, and the AI agent builds it atom by atom, solves its electronic structure in a tight-binding model, and posts the band structure in the chat. The example below has three turns: build a zigzag nanotube and classify it, build an armchair nanotube and compare, then sweep ten zigzag nanotubes to find which are metallic and how the gap depends on the diameter.

Why the Rolling Direction Decides

Graphene is a semiconductor with a gap of zero. Its valence band and conduction band touch at two points of the Brillouin zone, called K and K′, and are separated everywhere else. The electrons that carry the current are the ones in the p_z orbitals, one per carbon atom, standing perpendicular to the sheet. In the simplest model of graphene each electron hops between the p_z orbitals of nearest-neighbor carbon atoms with an energy of about 2.7 eV, and that single hopping energy produces the two bands and their touching points. This is the π tight-binding model, which the agent solves.

Rolling the sheet into a nanotube adds one condition. An electron wave traveling around the circumference must return to its starting value after one lap, so only wavelengths that fit a whole number of times around the nanotube are allowed. In the Brillouin zone of graphene this restricts the electron to a set of parallel lines, one for each allowed wavelength, and the integers (n, m) fix where the lines fall. When one of the lines passes through a K point, the nanotube inherits the touching bands of graphene and is metallic. When every line misses, the bands closest to K are separated by a gap, and the nanotube is a semiconductor. Whether a line passes through K depends on n − m. If n − m is a multiple of three, a line passes through. Otherwise the nearest line misses K by one third of the line spacing, and the gap is set by that distance. The line spacing is inversely proportional to the circumference, so the gap is inversely proportional to the diameter d. In the π model the gap of a semiconducting nanotube is 2·t·a_cc/d, where t is the hopping energy and a_cc the carbon–carbon bond length of 0.142 nm. With t = 2.7 eV, that is 0.77 eV·nm divided by the diameter. This way of obtaining the bands of a nanotube from the bands of graphene is called zone folding.

A Zigzag Nanotube and an Armchair Nanotube

In the first turn the agent builds a (10,0) zigzag nanotube, 0.78 nm across and 2.13 nm long, with 200 carbon atoms. It loads the segment into the lab and solves the infinitely long nanotube by repeating one period along the axis. The band-structure chart shows a gap of 0.95 eV at the center of the zone, and 10 − 0 leaves a remainder of 1 on division by three, so the nanotube is a semiconductor. In the second turn the agent builds a (6,6) armchair nanotube of the same length, 0.81 nm across, with 192 atoms. Its band-structure chart shows two bands crossing at the Fermi level, two thirds of the way from the zone center to the zone boundary, and no gap. The two nanotubes have nearly the same diameter and nearly the same number of atoms per nanometer of length, and differ in the direction in which the sheet was rolled.

Each turn also posts an energy-level chart for the 2 nm segment the lab holds, and the segment behaves differently from the infinite nanotube. The (10,0) segment has fourteen levels within 0.2 eV of the Fermi level, seven at each end, in the middle of what should be a 0.95 eV gap. They are localized on the two open rims, where every carbon atom has lost one of its three bonds. A zigzag edge of graphene carries a band of such states pinned at the Fermi level, as Nakada, Fujita, Dresselhaus and Dresselhaus showed in 1996, and a zigzag nanotube cut open has one such edge at each end. The agent tells the two kinds of level apart by where each wavefunction is concentrated, and the chart draws the rim states dashed. The extended states of the segment are 2.3 eV apart, far more than the 0.95 eV of the infinite nanotube, because a 2 nm box quantizes them. A real nanotube is capped or chemically terminated, so the rim states are absent and the nanotube is classified by the gap of the infinite one. The (6,6) segment shows a second finite-size effect. Its highest occupied and lowest empty levels are 0.59 eV apart, although the infinite nanotube is metallic, because a 2 nm piece of any metal has discrete levels. Armchair rims carry no edge states, so its level chart has no dashed lines.

Ten Zigzag Nanotubes

The third turn builds every zigzag nanotube from (7,0) to (16,0), solves each one, and plots the band gap against the diameter. Three of the ten have no gap, and they are the three whose n is divisible by three. The other seven have gaps that fall as the nanotube widens, from 1.33 eV at 0.55 nm to 0.60 eV at 1.25 nm, and the chart draws the zone-folding line 0.77 eV·nm/d through them. Every semiconducting point lies within five percent of the line.

The gaps alternate around the line. For (7,0), (10,0), (13,0) and (16,0) the nearest allowed line misses the K point on one side, and for (8,0), (11,0) and (14,0) on the other. The energy surface of graphene is a cone only close to K. Farther out it is warped into a triangle, so a line passing on one side of K gives a slightly smaller gap than a line at the same distance on the other side. A sweep of armchair nanotubes would put every point at zero, since n − m is zero for all of them.

Multiwall Nanotubes

The nanotubes Iijima reported in 1991 were multiwall: several nanotubes nested one inside the next, 0.34 nm apart, the spacing of the layers in graphite. The walls are held together by the same van der Waals attraction that holds the layers of graphite together, with no bonds between them, and the inner walls can slide out of the outer ones. Each wall is a nanotube with its own (n, m), so the rule applies wall by wall. Saito, Dresselhaus and Dresselhaus computed the bands of double-wall nanotubes in 1993 and found that the weak coupling between the walls leaves the character of each wall largely unchanged. Kwon and Tománek studied a (5,5) nanotube inside a (10,10) in 1998 and found that the coupling opens narrow pseudogaps near the Fermi level, which open and close as one wall moves relative to the other.

A second example, Multiwall Carbon Nanotubes, asks the same question of nested nanotubes. In the first turn the agent builds a (5,5) armchair nanotube 2 nm long inside a (10,10). The builder chooses the outer wall itself, as the armchair nanotube whose diameter is nearest 0.68 nm larger than the inner one, which places the walls 0.339 nm apart, with 480 carbon atoms in all. In the lab the carbon atoms of each wall are bonded by the Tersoff force field, and the two walls are held together by the interlayer attraction AIMS uses for graphite, so the walls keep their spacing but can slide along each other. The second turn asks whether the nanotube is metallic or semiconducting. The π model couples only bonded nearest neighbors, and atoms 0.34 nm apart on different walls are not bonded, so the agent solves each wall as a separate infinite nanotube and posts one band-structure chart per wall. Both armchair walls are metallic, as every armchair nanotube is. The same turn also builds a zigzag double-wall nanotube, (10,0) inside (19,0). Both of its walls are semiconducting, with gaps of 0.95 eV for the inner and 0.51 eV for the outer, and the wider outer wall sets the gap of the whole nanotube, since the first electrons to be excited across a gap are those of the wall with the smallest one. The last turn adds a third wall, (5,5)@(10,10)@(15,15), 2.03 nm across with 960 atoms, and every wall is metallic.

Whether a multiwall nanotube is metallic or semiconducting is therefore decided wall by wall, and the nanotube as a whole conducts if any of its walls is metallic. A nanotube with many walls almost always has at least one metallic wall, since one wall in three is metallic for walls grown at random. The outer walls of a real multiwall nanotube are also wide. By the inverse-diameter law, a semiconducting wall 10 nm across has a gap of about 0.08 eV, a few times the thermal energy at room temperature, so even the semiconducting walls of a wide nanotube carry current when it is warm. Frank, Poncharal, Wang and de Heer dipped multiwall nanotubes into liquid mercury in 1998 and measured their conductance in steps of one conductance quantum, carried by the outermost wall. Multiwall nanotubes therefore conduct, and they are the kind added to composites and battery electrodes to carry current. The two zigzag walls in the example show the other case: a narrow nanotube whose walls are all semiconducting is a semiconductor with the gap of its widest wall. The example's builder steps a zigzag wall from (n,0) to (n+9,0), which keeps the remainder of n on division by three, so a zigzag double-wall nanotube built in AIMS has two walls of the same kind. A semiconducting wall around a metallic one, which the rule allows, cannot yet be requested.

Running Experiments by Asking

The student names a nanotube by its two integers and asks whether it is metallic or semiconducting. The agent picks the tool, builds the structure, solves it, and posts the chart in the chat. Building a (6,6) nanotube atom by atom, setting up the repeated period, and telling rim states from extended states are the agent's work.

The Model and Its Limits

The agent builds each nanotube by geometry alone, from the two integers and the carbon–carbon bond length. The electronic structure comes from the nearest-neighbor π tight-binding model with a hopping energy of 2.7 eV, which is the model in which the (n, m) rule was first derived. The infinite nanotube is solved by applying Bloch's theorem along the axis to one period of the structure. The finite segment in the lab is solved as a cluster, every p_z orbital at once. The segment also carries the Tersoff carbon force field, so it can be heated, stretched or cut in the lab. The force field governs the motion of the atoms, and the tight-binding model governs the electrons.

The model treats the bonds of a rolled sheet exactly as it treats the bonds of a flat one. In a real nanotube the curvature mixes the in-plane σ orbitals into the π states, and for the nominally metallic zigzag nanotubes that mixing opens a small gap. Ouyang, Huang, Cheung and Lieber measured those gaps by tunneling spectroscopy in 2001 and found them to shrink as the inverse square of the nanotube radius. Armchair nanotubes have no such gap, because their symmetry protects the crossing. The (9,0), (12,0) and (15,0) nanotubes in the table are therefore semiconductors with small gaps where the model says zero. The same mixing lowers the gaps of the narrowest semiconducting nanotubes below the π-model values, so the (7,0) and (8,0) entries in the table are the least reliable. The model also omits the interaction between electrons, which matters for the ribbons in the Go further list. The hopping energy and the bond length are the two quantities in the model taken from experiment, and the tunneling measurements of 1998 gave the inverse-diameter law with a hopping energy close to the one used here.

Conclusion

This article uses the electronic structure of carbon nanotubes to illustrate a new way to learn science: students converse with an AI agent that turns a plain-language question into a quantum mechanical calculation and hands back the result as a chart. The rule that a nanotube is metallic when n − m is a multiple of three, normally a sentence in a nanoscience course, becomes something students test for themselves. They build a zigzag nanotube and an armchair nanotube of the same size, find a gap in one and none in the other, and sweep a family of ten to watch the gap fall with the diameter and vanish at every third member. They then nest one nanotube inside another and find that the rule holds wall by wall, so that a multiwall nanotube conducts whenever one of its walls is metallic. They also learn how a finite segment differs from an infinite nanotube, and where the model departs from the measured nanotubes. By pairing a physically grounded solver with an AI that can build the structures and run it, AIMS lets students investigate a rule they would otherwise only read about.

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