Boltzmann equation Handout

# Boltzmann equation Handout - University of California...

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University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor Boltzmann equation Handout.fm Transport Theory (read Lundstrom 3.1 - 3.4) Aim Develop a general approach for relating microscopic description of carrier motion to mac- roscopic description. Drift-Diffusion equation: Classical approach to device design Device behavior described by independent specification of material and structure. Modern devices μ , D: can’t bury microphysics (i.e. ballistic transport) also depend explicitly on device structure (i.e. nonlocal transport) Carrier energy & momentum distributions vary strongly in devices Carrier scattering : key to transport. Paradox: no scattering --> net J=0! consider simple band E = E o - E o cos( ka) material properties: μ , D + Boundary conditions device structure microphysics is buried in here k E v 1 h -- E K ------ E o a h --------- ka () sin ==

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- 93 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor periodic motion - Bloch oscillation. This has been experimentally observed in superlattices [“Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor super- lattice,” C. Waschke, et al., Phys. Rev. Lett. 70, 3319 - 3322 (1993). PDF copy is posted on the class web site.] Electrons oscillate through BZ [Bragg diffraction] localized in real space no net cur- rent ! Scattering damps oscillation resulting in a net motion in an applied field. Yet the motion is damped by scattering. The message is that transport depends on the balance between the applied driving force and dissipation by scattering forces . Boltzmann transport equation Semi-classical approach. In semiconductors, all the QM is buried in m*. Electron motion will be described by classical mechanics, except that scattering probabilities will also be derived using quantum mechanics. Complete description: could solve Newton equations for each particle (electron): for all . This is clearly not feasible. Instead, we take a statistical approach: define distribution function (single particle): : probability of finding a particle at , with momentum at time t. “phase space”: 6 dimensional space - x, y, z, p x , p y , p z The particle density is found as: Since momentum is discrete, we show a summation over allowed values of p. However, it is dp i dt ------- qE F rpt ,, () + = random forces due to phonons, impurities i 1 N = frpt r p
- 94 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor more convenient to convert to an appropriate integral.

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## This note was uploaded on 08/01/2008 for the course EE 230 taught by Professor Bokor during the Spring '08 term at Berkeley.

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Boltzmann equation Handout - University of California...

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