2D companion to the 3D rendered scene: the identical Landauer-Büttiker transport physics, read off a top-down electron-flow lane and a resistance-vs-length curve instead of a rotatable tube bundle. A defect-free metallic single-wall carbon nanotube carries current through exactly 4 quantum-mechanical conduction channels (2 linear subbands, each already spin-degenerate). For a bundle of M parallel tubes of length L:
R(L) = (h / 4e²N) × (1 + L/λ)
N = 4M h/4e² ≈ 6.453 kΩ (quantum resistance per channel)
λ = electron mean free path (acoustic-phonon-limited, ~1 μm at 300 K)
When L ≪ λ, transport is ballistic: electrons cross the tube without scattering and R collapses to a length-independent quantum contact resistance R_c = (h/4e²)/N. As L grows past λ, scattering events accumulate and transport turns diffusive: R grows linearly with L, exactly like an Ohmic wire, with slope (h/4e²)/(Nλ). The flow lane flashes red at each scattering event so the crossover is visible, and the curve panel plots the full R(L) shape for the current λ and bundle size with the current setting marked.
- Length L — physical span of the interconnect between contacts.
- Mean free path λ — average distance between phonon-scattering events; shrinks with more defects or higher temperature.
- Bundle count M — parallel metallic tubes; each adds 4 channels, so contact resistance drops as 1/M.
Why it matters: this is exactly the tradeoff nanoelectronics interconnect design faces as wires shrink below a micron — copper's resistivity rises sharply from surface scattering at these dimensions, while a ballistic CNT bundle instead hits a resistance floor set by quantum contacts, which is why CNT and graphene interconnects are studied as copper's replacement for the shortest on-chip wires.