Abstract
The vibrational properties of crystalline bulk materials are well described by Debye theory, which successfully predicts the quadratic ω(2) low-frequency scaling of the vibrational density of states. However, the analogous framework for nanoconfined materials with fewer degrees of freedom has been far less well explored. Using inelastic neutron scattering, we characterize the vibrational density of states of amorphous ice confined inside graphene oxide membranes and we observe a crossover from the Debye ω(2) scaling to an anomalous ω(3) behaviour upon reducing the confinement size L. Additionally, using molecular dynamics simulations, we confirm the experimental findings and prove that such a scaling appears in both crystalline and amorphous solids under slab-confinement. We theoretically demonstrate that this low-frequency ω(3) law results from the geometric constraints on the momentum phase space induced by confinement along one spatial direction. Finally, we predict that the Debye scaling reappears at a characteristic frequency ω(×) = vL/2π, with v the speed of sound of the material, and we confirm this quantitative estimate with simulations.