Recent electron microscopy experiments that use focused electron beams as nanoscale heat sources or thermometers have the potential to improve the understanding of heat flow at single-digit nanometer lengthscales. However, the potential effects of electron-beam induced heating in these experiments has not been previously studied from a sub-continuum perspective. Altough Fourier's law predicts that the beam-inducted heating will be negligibly small in high thermal conductivity materials, previous optical measurements have indicated that Fourier's law underpredicts the temperature rise when the characteristic heat source size (here the electron beam radius) is smaller than the heat carrier mean free path. Indeed, the most common beam radii used in scanning transmission electron microscopy or scanning electron microscopy experiments are \<10 nm, which is smaller than the dominant phonon mean free paths in many crystalline semiconductors at room temperature. Because the beam radii are smaller than the mean free path, the true magnitude of electron beam-induced heating in electron microscopy experiments cannot be predicted using existing models based on continuum heat conduction.

Here, beam heating in nonmetallic samples is modeled by applying a recently developed general solution of the Boltzmann Transport Equation (BTE) under the relaxation time approximation \[Hua and Minnich, PRB 90.21 (2014): 214306\] to the electron-beam heating geometry. The BTE solution is valid for an isotropic material in an infinite domain undergoing time-periodic heating from a cylindrical Gaussian heat source; in this model, the thermal penetration depth is analagous to the outer radius of the sample. The inputs to the BTE model are the phonon dispersion relations and bulk scattering rates, which are determined by fitting commonly used analytical expressions to experimental data. These BTE results show that ballistic phonon effects in this radial heat spreading scenario are conveniently represented using a ballistic thermal resistance, rather than the more traditionally-utilized effective thermal conductivity. This ballistic resistance has previously been used to interpret experimental data in optical experiments. Calculations of this ballistic resistance for Si, GaAs, and 3C-SiC show that ballistic effects dominate the total thermal resistance for typical beam radii (\<10 nm), indicating that the ballistic resistance could be measured using electron beam heating experiments. However, the magnitude of the temperature rise remains small (\<1 K), even when considering these ballistic effects, indicating that the beam-induced heating will likely not pose a major challenge to nanoscale temperature mapping measurements in practice. In summary, 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.  
  
Ref.: Wehmeyer, Journal of Applied Physics 126, 124306 (2019).

\*Welch Foundation (Grant #C-2023)


Boltzmann Equation Modeling of Quasiballistic Phonon Transport From a Cylindrical Electron Beam Heat Source

Division / Sponsors: 
Heat Transfer Division (HTD), Fluids Engineering Division (FED)

Heat Transfer Summer Conference
Paper Proceedings: https://asmedigitalcollection.asme.org/HT/HT2020/volume/83709

18th International Conference on Nanochannels, Microchannels, and Minichannels
Paper Proceedings: https://asmedigitalcollection.asme.org/ICNMM/ICNMM2020/volume/83693

Fluids Engineering Division Summer Meeting
Paper Proceedings: https://asmedigitalcollection.asme.org/FEDSM/FEDSM2020/volume/83716

HT-ICNMM2020 - Virtual Conference, SHTC - Virtual Conference, FEDSM2020 - Virtual Conference

Supporting the need among researchers worldwide for additional online and multimedia content, ASME now offers the ASME Conference Video Collection. This platform will collect and provide long-term access to conference presentation videos, encompassing all technology areas within ASME's technical conference and events program. About 30,000-40,000 engineers across all mechanical engineering and related disciplines participate in ASME conferences each year. This includes authors of approximately 6,000 to 8,000 technical papers published by ASME, the majority of whom also submit video presentations for our virtual conferences. The video collection will be available as an add-on to existing product packages or as a separate subscription.