Boltzmann Transport Equation for Particle Transport Distribution Function
Boltzmann Transport Equation for Particle Transport Distribution Function of Particles: f = f(r, p, t) --probability of particle occupation of momentum p at location r and time t Equilibrium Distribution: f 0, i. e. Fermi-Dirac for electrons, Bose-Einstein for phonons Non-equilibrium, e. g. in a high electric field or temperature gradient: Relaxation Time Approximation Relaxation time t
Energy Flux q Energy flux in terms of particle flux carrying energy: v dk q k f Vector Integrate over all the solid angle: Scalar Integrate over energy instead of momentum: Density of States: # of phonon modes per frequency range
Continuum Case BTE Solution: Quasi-equilibrium Energy Flux: Direction x is chosen to in the direction of q Fourier Law of Heat Conduction: t(e) can be treated using Callaway method (Phys. Rev. 113, 1046) If v and t are independent of particle energy, e, then Kinetic theory:
At Small Length/Time Scale (L~l or t~t) Define phonon intensity: From BTE: 0 Equation of Phonon Radiative Transfer (EPRT) (Majumdar, JHT 115, 7): Heat flux: Acoustically Thin Limit (L<<l) and for T << q. D Acoustically Thick Limit (L>>l)
Outline ü Macroscopic Thermal Transport Theory – Diffusion -- Fourier’s Law -- Diffusion Equation ü Microscale Thermal Transport Theory – Particle Transport -- Kinetic Theory of Gases -- Electrons in Metals -- Phonons in Insulators -- Boltzmann Transport Theory Thermal Properties of Nanostructures -- Thin Films and Superlattices -- Nanowires and Nanotubes -- Nano Electromechanical System 5
Thin Film Thermal Conductivity Measurement 3 w method (Cahill, Rev. Sci. Instrum. 61, 802) Metal line L Substrate Thin Film 2 b I 0 sin(wt) V • I ~ 1 w • T ~ I 2 ~ 2 w • R ~ T ~ 2 w • V~ IR ~3 w 6
Silicon on Insulator (SOI) Ju and Goodson, APL 74, 3005 IBM SOI Chip Lines: BTE results Hot spots! 7
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