The effect of confinement on the acoustic phonon dispersion relation and heat capacity in free-standing silicon membranes is investigated, with thickness values down ∼ 8 nm. The discrete phonon branches are observed by angle-resolved inelastic light scattering spectroscopy. The fundamental flexural mode was observed to have a scattering intensity nearly two orders of magnitude larger than the fundamental dilatational mode, which is ascribed to its large out-of-plane density of states and quadratic dispersion. The quadratic dispersion also results in a reduction of the phase and group velocities of the fundamental flexural mode by more than one order of magnitude compared to bulk values. To investigate the effect of this behavior on the thermal conductivity, we perform calculations based on continuum elasticity theory to estimate corresponding changes in the heat capacity. This work provides a basis to investigate the effects of the frequency and dimension dependence of other phonon properties, in particular in the sub-20 nm regime, where phonon-phonon relaxation times, density of states and thermal conductivity are expected to possess different spectral dependencies than for bulk materials.
Discrete and Weyl density of states for photonic dispersion relation
