On the electronic and vibrational dimensionality of nanometer-scale silicon structures

Abstract

We discuss the problem of assessing the electronic and vibrational dimensionality of a semiconductor nanostructure: How thin and/or wide must a nanostructure be in order to induce electron and phonon confinement? Clarifying the physical justification for common criteria found in the literature, we view the electron coherence length (defined as the electron and phonon inelastic mean free path) as their `field of view' and argue (or, better yet, `speculate') that this sets the important length scale. Considering the example of Si nanosheets at room temperature, and drawing from results found in the literature, we estimate that the critical length below which electrons are subject to quantum confinement is of the order of (or smaller than) 8 nm, when their coherence length is determined by energy losses to phonons and remote phonons in gated structures. On the contrary, no single length-scale can be given for phonons: Taking their coherence length as determined by scattering with electrons and anharmonic three-phonon processes, short wavelength acoustic and optical phonons may be confined only by structures as small as 10 nm. Long-wavelength acoustic phonons, instead, may exhibit a coherence length of the order of 1 micrometer, so that they may be confined over much larger distances.