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MICRO/NANOSCALE HEAT TRANSFER—PART II

Contribution of Ballistic Electron Transport to Energy Transfer During Electron-Phonon Nonequilibrium in Thin Metal Films

[+] Author and Article Information
Patrick E. Hopkins1

Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904-4746

Pamela M. Norris2

Department of Mechanical and Aerospace Engineering, University of Virginia, Charlottesville, VA 22904-4746pamela@virginia.edu

1

Present address: Engineering Sciences Center, Sandia National Laboratories, P.O. Box 5800, Albuquerque, NM 87185-0346.

2

Corresponding author.

J. Heat Transfer 131(4), 043208 (Feb 20, 2009) (8 pages) doi:10.1115/1.3072929 History: Received June 15, 2008; Revised September 10, 2008; Published February 20, 2009

With the ever decreasing characteristic lengths of nanomaterials, nonequilibrium electron-phonon scattering can be affected by additional scattering processes at the interface of two materials. Electron-interface scattering would lead to another path of energy flow for the high-energy electrons other than electron-phonon coupling in a single material. Traditionally, electron-phonon coupling in transport is analyzed with a diffusion (Fourier) based model, such as the two temperature model (TTM). However, in thin films with thicknesses less than the electron mean free path, ballistic electron transport could lead to electron-interface scattering, which is not taken into account in the TTM. The ballistic component of electron transport, leading to electron-interface scattering during ultrashort pulsed laser heating, is studied here by a ballistic-diffusive approximation of the Boltzmann transport equation. The results for electron-phonon equilibration times are compared with calculations with TTM based approximations and experimental data on Au thin films.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Schematic of temperature changes of electron and lattice systems immediately after short-pulsed laser heating, in the case when the film thickness is less than the ballistic penetration depth. (a) Insulated boundary between the film and substrate: electron-phonon coupling will dominate the electron scattering events and drive electron cooling. Although electrons will traverse to the film/substrate interface, assuming an insulated boundary, the electrons will elastically reflect off the interface and travel back into the film and scatter with phonons in the film. (b) Uninsulated boundary between the film and substrate; electrons can inelastically scatter at the interface creating another form of energy loss from the electron system in addition to electron-phonon scattering in the film.

Grahic Jump Location
Figure 2

Average power density transferred from the electron system assuming no energy loss to the substrate (elastic electron-interface scattering) as a function of maximum electron temperature in the (a) ballistic regime assuming τee of 50 fs, 200 fs, 350 fs, and 500 fs and τep of 4 ps (Eq. 20) and (b) diffusive regime (electron-phonon coupling) assuming τee of 200 fs and τep of 2 ps, 4 ps, 6 ps, and 8 ps (Eq. 21).

Grahic Jump Location
Figure 3

Ratio of average power density transferred from ballistic processes to diffusive processes (ratio of Eq. 20 to Eq. 22) as a function of maximum electron temperature. These calculations assume a temperature dependent electron-phonon relaxation time of τep=γTe,max/G and no energy loss to the substrate. Calculations shown assume τee of 50 fs, 200 fs, 350 fs, and 500 fs.

Grahic Jump Location
Figure 4

Average power density transferred from the electron system assuming energy loss to the substrate (inelastic electron-interface scattering) as a function of maximum electron temperature in the (a) ballistic regime assuming τei of 15 fs, 50 fs, 200 fs, 350 fs, and 500 fs and τep of 4 ps (Eq. 30) and (b) diffusive regime (electron-phonon coupling) assuming τee of 200 fs and τep of 2 ps, 4 ps, 6 ps, and 8 ps (Eq. 31).

Grahic Jump Location
Figure 5

Ratio of average power density transferred from ballistic processes to diffusive processes (ratio of Eq. 30 to Eq. 32) as a function of maximum electron temperature. These calculations assume a temperature dependent electron-phonon relaxation time of τep=γTe,max/G and energy loss to the substrate from inelastic electron-interface scattering events during electron-electron thermalization. Calculations shown assume τei of 15 fs, 50 fs, 200 fs, 350 fs, and 500 fs. Also shown in the figure are experimental pump probe Gs measured on 20 nm Au films on Si substrates as a function of maximum electron temperature as measured by Hopkins and Norris (14). Their data are normalized by the accepted value of G in Au, G=2.4×1016 W m−3 K−1. The experimental work from Hopkins and Norris concluded that electron-interface scattering could increase the rate of electron-phonon equilibration in the film by providing another channel of energy transfer. This same conclusion is apparent in the calculations shown in this figure.

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