Ten years of carbon nanotube simulations, in one plot

Between 2002 and 2012 I ran, by a rough count, several thousand tight-binding molecular dynamics simulations of single-walled carbon nanotubes. Almost all of them did the same thing: take a tube, pull on it, and watch what happens to its electronic structure.

The method, briefly#

Tight-binding molecular dynamics sits in an awkward but useful place. Ab initio DFT is more accurate and hopelessly expensive at the system sizes we needed; classical potentials are cheap but cannot tell you anything about a band gap. TBMD keeps a quantum-mechanical description of the electrons while remaining tractable for a few thousand atoms.

The catch is that diagonalising the Hamiltonian is $O(N^3)$, which is fatal. Order-$N$ methods get around this by exploiting the locality of the density matrix — the observation that, in a system with a band gap, the density matrix elements decay exponentially with distance:

$$\rho_{ij} \sim e^{-\gamma |r_i - r_j|}$$

Truncate below a cutoff and the cost collapses to linear. That is what made a decade of this work possible on the hardware we had.

The finding that held up#

Across chiralities, temperatures and defect densities, one relationship kept reappearing: band gap responds almost linearly to axial strain, and the sign of the slope depends on chirality mod 3.

For a zigzag $(n, 0)$ tube:

• $n \bmod 3 = 1$ → the gap opens under tension
• $n \bmod 3 = 2$ → the gap closes under tension
• $n \bmod 3 = 0$ → metallic, and a small gap opens either way

This is not our discovery — it falls out of zone-folding arguments — but seeing it emerge from an atomistic simulation that knows nothing about zone folding is the kind of thing that makes you trust your code.

Where the defects came in#

Perfect nanotubes are a physicist's fiction. Real ones have vacancies. So we started removing atoms, and the picture got more interesting: a single vacancy introduces localised states in the gap, and at sufficient vacancy density the strain response stops being linear altogether.

The practical consequence is uncomfortable for anyone hoping to build a strain sensor out of these things: your calibration curve depends on your defect density, which you generally do not know.

What I took with me#

I do not simulate nanotubes any more. But two habits from that decade have followed me into machine learning:

1. Know what your model is allowed to assume. Order-$N$ methods work because of a physical property (density-matrix locality). Use them where that property does not hold and they will quietly give you garbage. Neural networks have exactly the same failure mode, minus the honesty of a physical justification.

2. Cheap and wrong beats expensive and wrong. The whole reason to do TBMD is that you have decided which errors you can live with. Most modelling is that decision, made well or badly.