Modeling ballistic phonon transport from a cylindrical electron beam heat source

Abstract

Recent electron microscopy experiments have used focused electron beams as nanoscale heat sources or thermometers to enable high spatial resolution studies of heat transfer in nanostructures. When the electron beam radius is smaller than the heat carrier mean free path, Fourier’s law will underpredict the temperature rise due to electron beam-induced heating, motivating the development of subcontinuum models to interpret thermal electron microscopy measurements. Here, electron beam-induced heating of nonmetallic samples is modeled by applying a recently developed general solution of the governing Boltzmann transport equation (BTE) under the relaxation time approximation. The analytical BTE solution describes thermal phonon transport from a time-periodically heated cylindrical region in a homogeneous infinite medium. The BTE results show that ballistic phonon effects in this radial heat spreading scenario are more conveniently represented using a ballistic thermal resistance rather than an effective thermal conductivity. Calculations of this ballistic resistance for three semiconductors (Si, GaAs, and 3C-SiC) show that ballistic effects dominate the total thermal resistance to radial heat flow for typical STEM or SEM beam radii (<10 nm), indicating that the ballistic resistance could potentially be measured using thin-film electron beam heating experiments. However, combining the BTE solution with recent calorimetric measurements shows that the magnitude of the temperature rise remains negligibly small (<1 K) under typical electron microscopy conditions, even when considering these ballistic effects. These BTE modeling results can be used to quantify electron beam-induced heating or to design experiments probing ballistic phonon transport using electron beam heat sources.