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: f0, 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 t Relaxation time

Energy Flux Energy flux in terms of particle flux carrying energy: q v Energy flux in terms of particle flux carrying energy: 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 Direction x is chosen to in the direction of q Energy Flux: 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: Equation of Phonon Radiative Transfer (EPRT) (Majumdar, JHT 115, 7): Heat flux: Acoustically Thin Limit (L<<l) and for T << qD Acoustically Thick Limit (L>>l)

Outline Macroscopic Thermal Transport Theory – Diffusion -- Fourier’s Law -- Diffusion Equation 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

Thin Film Thermal Conductivity Measurement 3w method (Cahill, Rev. Sci. Instrum. 61, 802) Metal line Thin Film L 2b V I ~ 1w T ~ I2 ~ 2w R ~ T ~ 2w V~ IR ~3w I0 sin(wt) Substrate

Silicon on Insulator (SOI) Ju and Goodson, APL 74, 3005 IBM SOI Chip Lines: BTE results Hot spots!