1. 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

2. 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

3. 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:

4. 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)

5. 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

6. 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

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