Excess speicific heat of the gapped sliding phonons in the incommensurate composite crystal Sr$_{14}$Cu$_{24}$O$_{41}$

We show that low temperature specific heat (C$_p$) of the incommensurate chain-ladder system Sr$_{14}$Cu$_{24}$O$_{41}$ is enriched by the presence of a rather large excess contribution of non-magnetic origin. Diluted Al doping at the Cu site or annealing the crystal in an O$_2$ atmosphere suppresses this feature considerably. Using the THz time-domain spectroscopy, we show that the occurrence of excess specific heat is associated with the presence of very low-energy ($\sim$ 1 meV) gapped phonon modes that originate due to the sliding motion of oppositely charged mutually incommensurate chain and ladder layers. ; Comment: 5pages, 4figures