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A5: Nanoscale and low dimensional effects
Steady-state and transient phonon transport under ballistic, quasi-ballistic, and diffusive conditions is important for modeling thermal transport in thermoelectrics, nanoelectronic devices, thin dielectric films, and in measurements such as time domain thermal reflectivity. It is generally understood that Fourier’s Law cannot describe steady-state heat transport under short spatial scales and that the hyperbolic heat equation (Cattaneo equation) cannot describe transient heat conduction in the ballistic limit [1]. To address these problems, several methods have been developed. We show that steady-state and transient heat conduction can be described from the ballistic to diffusive regimes by conventional diffusion equations – if the proper boundary conditions are used. This conclusion follows from the Boltzmann Transport Equation.
We begin with a simple version of the Boltzmann equation – the 1-flux equations of McKelvey [2] and show that the equations can be re-written without approximation as Fourier’s Law of heat conduction [3]. We show that the temperature jumps discussed in [1] can be obtained exactly from a solution to Fourier’s Law along with the Casimir limit for ballistic transport. We then discuss the extension to time-dependent problems. It is possible to show how the full Boltzmann equation can be written as a multi-dimensional Fourier’s Law in phase space.
In summary, we show that conventional diffusion equations describe phonon transport from the ballistic to diffusive regimes when boundary conditions are correctly specified.
[1] A. A. Joshi and A. Majumdar, “Transient ballistic and diffusive phonon heat transport in thin films,” J. Appl. Phys., 74, 31, 1993.
[2] J. P. McKelvey, R. L. Longini, and T. P. Brody, “Alternative approach to the solution of added carrier transport problems in semiconductors,” Phys. Rev.,123, 51, 1961.
[3] W. Shockley, “Diffusion and drift of minority carriers in semiconductors for comparable capture and scattering mean free paths,” Phys. Rev., 125, 1570, 1962.