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Moir\'e patterns are additional, long-range periodicities in twistedcrystalline bilayers. They are known to fundamentally change the electronicstates of the layers, but similar effects on their mechanical and vibrationalproperties have not been discussed so far. Here we show that the Moir\'epotential shifts the radial breathing mode in double walled carbon nanotubes(DWCNTs). The change of frequency is expected to be proportional to the shiftin optical transition energies, which are induced by the Moir\'e patterns. Toverify our model we performed resonant Raman scattering on purified and sortedsemiconducting DWCNTs. We find that the radial breathing mode shifts up to 14cm$^{-1}$ higher in energy followed by optical transitions energiesdisplacement up to 200 meV to lower energies, compared to the single-walledtubes. We show how to identify the strong coupling condition in DWCNTs fromtheir phonon frequencies and construct a Kataura plot to aid their futureexperimental assignment.

Moir\'e-induced Vibrational Coupling in Double Walled Carbon Nanotubes