Thermal transport at the nanoscale: A Fourier's law vs. phonon Boltzmann equation study

Abstract

Steady-state thermal transport in nanostructures with dimensions comparable to the phonon mean-free-path is examined. Both the case of contacts at different temperatures with no internal heat generation and contacts at the same temperature with internal heat generation are considered. Fourier's Law results are compared to finite volume method solutions of the phonon Boltzmann equation in the gray approximation. When the boundary conditions are properly specified, results obtained using Fourier's Law without modifying the bulk thermal conductivity are in essentially exact quantitative agreement with the phonon Boltzmann equation in the ballistic and diffusive limits. The errors between these two limits are examined in this paper. For the four cases examined, the error in the apparent thermal conductivity as deduced from a correct application of Fourier's Law is less than 6%. We also find that the Fourier's Law results presented here are nearly identical to those obtained from a widely used ballistic-diffusive approach, but analytically much simpler. Although limited to steady-state conditions with spatial variations in one dimension and to a gray model of phonon transport, the results show that Fourier's Law can be used for linear transport from the diffusive to the ballistic limit. The results also contribute to an understanding of how heat transport at the nanoscale can be understood in terms of the conceptual framework that has been established for electron transport at the nanoscale.

Study of Heat Dissipation Mechanism in Nanoscale MOSFETs Using BDE Model

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“…Due to the miniaturization the thermal conductivity, reduce by scattering [24,25]. The scattering mechanism induced to the use of the ETC [26][27][28][29]. We have proposed a theoretical approach, which describe the nature of phonon collision with boundary.…”

Modeling quasi-ballistic transient thermal transport with spatially sinusoidal heating: A McKelvey-Shockley flux approach

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“…The McK-S framework can be viewed as a simple version of the rigorous BTE, which has the benefits of physical transparency and computational efficiency, and could prove useful for treating complicated problems. Interestingly, the McK-S equations were shown to be mathematically equivalent to well-known diffusion equations [27][28][29][30], which when solved with the appropriate physical boundary conditions provide surprisingly good agreement with more rigorous approaches. It is important to explore the accuracy of McK-S under different conditions, by benchmarking this approach against the BTE, to understand its limits and provide improvements.…”

Non-Equilibrium Thermal Transport: A Review of Applications and Simulation Approaches

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