  Interlayer van der Waals (vdW) coupling is generic in two-dimensional
materials such as graphene and transition metal dichalcogenides, which can
induce very low-energy phonon modes. Using high-resolution inelastic hard x-ray
scattering, we uncover the ultra-low energy phonon mode along the Cu-O bond
direction in the high-$T_c$ cuprate (Bi,Pb)$_2$(Sr,La)$_2$CuO$_{6+\delta}$
(Bi2201). This mode is independent of temperature, while its intensity
decreases with doping in accordance with an increasing c-axis lattice
parameter. We compare the experimental results to first-principles density
functional theory simulations and identify the observed mode as a van der Waals
phonon, which arises from the shear motion of the adjacent Bi-O layers. This
shows that Bi-based cuprate has similar vibrational properties as graphene and
transition metal dichalcogenides, which can be exploited to engineer novel
heterostructures.
