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Research Papers: Micro/Nanoscale Heat Transfer

Transient Energy and Heat Transport in Metals: Effect of the Discrete Character of the Lattice

[+] Author and Article Information
Y. Ezzahri

Département Fluides, Thermique, Combustion, Institut Pprime, ENSIP-Bâtiment de mécanique, CNRS-Université de Poitiers-ENSMA, 2 rue Pierre Brousse, Poitiers Cedex F 86022, Franceyounes.ezzahri@univ-poitiers.fr

K. Joulain

Département Fluides, Thermique, Combustion, Institut Pprime, ENSIP-Bâtiment de mécanique, CNRS-Université de Poitiers-ENSMA, 2 rue Pierre Brousse, Poitiers Cedex F 86022, France

A. Shakouri

Department of Electrical Engineering, University of California Santa Cruz, 1156 High Street, Santa Cruz, CA 95064

J. Heat Transfer 133(7), 072401 (Apr 05, 2011) (14 pages) doi:10.1115/1.4003577 History: Received October 12, 2010; Revised January 31, 2011; Published April 05, 2011; Online April 05, 2011

A recently developed Shastry’s formalism for energy transport is used to analyze the temporal and spatial behaviors of the electron energy and heat transport in metals under delta function excitation at the surface. Comparison with Cattaneo’s model is performed. Both models show the transition between nonthermal (ballistic) and thermal (ballistic-diffusive) regimes. Furthermore, because the new model considers the discrete character of the lattice, it highlights some new phenomena, such as damped oscillations, in the energy transport both in time and in space. The energy relaxation of the conduction band electrons in metals is considered to be governed by the electron-phonon scattering, and the scattering time is taken to be averaged over the Fermi surface. Using the new formalism, one can quantify the transfer from nonthermal modes to thermal ones as energy propagates in the material and it is transformed into heat. While the thermal contribution shows a wave-front and an almost exponentially decaying behavior with time, the nonthermal part shows a wave-front and a damped oscillating behavior. Two superimposed oscillations are identified, a fast oscillation that is attributed to the nonthermal nature of energy transport at very short time scales and a slow oscillation that describes the nature of the transition from the nonthermal regime to the thermal regime of energy transport.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic diagram of the metal being excited by a laser delta pulse at its free top surface (x=0)

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Figure 2

Temporal behavior of the (a) thermal (Eq. 8) and (b) nonthermal (Eq. 8) parts of the electron energy density Green’s function in Shastry’s model at the top free surface of gold and aluminum at room temperature. The inset in (b) shows a zoom of the behavior of the nonthermal contribution at long time scale for gold. (c) Comparison between the thermal contributions (dashed line), the sum of the nonthermal and thermal contributions (solid line) to the total electron energy density Green’s function at the top free surface of gold at room temperature, with Fourier’s model (dotted line). (d) Comparison between the temporal behaviors of the nonthermal contribution to the total electron energy density Green’s function at the top free surface of gold at room temperature, as calculated based on Shastry’s model (solid line) and Cattaneo’s model (dashed line in the inset).

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Figure 3

[(a) and (b)] Frequency behavior of the thermal contribution, [(c) and (d)] the nonthermal contribution, and [(e) and (f)] the total electron energy density Green’s function at the top free surface of gold (solid line) and aluminum (dashed line) at room temperature in both [(a), (c), and (e)] Cattaneo’s model and [(b), (d), and (f)] Shastry’s model

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Figure 4

Frequency behavior of (a) the thermal contribution, (b) the nonthermal contribution, and (c) the total electron energy density Green’s function at the top free surface of gold at room temperature in both Cattaneo’s model (solid line) and Shastry’s model (solid-dashed line). The dashed line in (c) describes Fourier’s model.

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Figure 5

[(a) and (b)] Frequency behavior of the thermal contribution, [(c) and (d)] the nonthermal contribution, and [(e) and (f)] the total electron energy density Green’s function at the top free surface of gold at different temperatures in both [(a), (c), and (e)] Cattaneo’s model and [(b), (d), and (f)] Shastry’s model

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Figure 6

Comparison between the behaviors of the thermal contribution to the total electron energy density Green’s function of gold at room temperature, as calculated based on both Shastry and Cattaneo models (solid line) with Fourier’s model (dashed line), (a) temporal behavior at different locations y and (b) spatial behavior at different times η. C, F, and S refer to Cattaneo, Fourier, and Shastry, respectively.

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Figure 7

[(a) and (b)] Temporal and [(c) and (d)] spatial behaviors of the nonthermal contribution to the total electron energy density Green’s function of gold at room temperature, as calculated using [(a) and (c)] Shastry’s model and [(b) and (d)] Cattaneo’s model. The insets in [(a) and (b)] show a zoom of the slow oscillations while the insets in [(c) and (d)] show the spatial behavior of the nonthermal contribution at η=30.

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Figure 8

(a) Temporal behavior of the total electron energy density Green’s function of gold at room temperature at different locations y, as calculated based on Shastry’s model (solid line), in comparison with full Cattaneo’s model (dashed line) and Fourier’s model (dotted line). (b) Temporal behavior of the total electron energy density Green’s function of gold at room temperature at different locations y, as calculated based on approximated Cattaneo’s model (solid line), in comparison with full Cattaneo’s model (dashed line) and Fourier’s model (dotted line). C, F, and S refer to Cattaneo, Fourier, and Shastry, respectively.

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Figure 9

(a) Spatial behavior of the total electron energy density Green’s function of gold at room temperature at different times η, as calculated based on Shastry’s model (solid line), in comparison with full Cattaneo’s model (dashed line) and Fourier’s model (dotted line). (b) Spatial behavior of the total electron energy density Green’s function of gold at room temperature at different times η, as calculated based on approximated Cattaneo’s model (solid line), in comparison with full Cattaneo’s model (dashed line) and Fourier’s model (dotted line). C, F, and S refer to Cattaneo, Fourier, and Shastry, respectively.

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