Probing the phonon confinement in ultrasmall silicon nanocrystals reveals a size-dependent surface energy

We validate for the first time the phenomenological phonon confinement model (PCM) of H. Richter, Z. P. Wang, and L. Ley [Solid State Commun. 39, 625 (1981)] for silicon nanostructures on the sub-3 nm length scale. By invoking a PCM that incorporates the measured size distribution, as determined from cross-sectional transmission electron microscopy (X-TEM) images, we are able to accurately replicate the measured Raman line shape, which gives physical meaning to its evolution with high temperature annealing and removes the uncertainty in determining the confining length scale. The ability of our model to explain the presence of a background scattering spectrum implies the existence of a secondary population of extremely small (sub-nm), amorphous silicon nanoclusters which are not visible in the X-TEM images. Furthermore, the inclusion of an additional fitting parameter, which takes into account the observed peak shift, can be explained by a size-dependent interfacial stress that is minimized by the nanocluster/crystal growth. From this we obtain incidental, yet accurate estimates for the silicon surface energy and a Tolman length, δ ≈ 0.15 ± 0.1 nm using the Laplace-Young relation.