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Transcript
1 Boltzmann Transport Equation LECTURE 2 • Boltzmann Transport Equation • Continuity (Balance) Equations • Method of Moments 2 Sec. 5.2 Boltzmann Transport Equation • This famous equation describes how an electron moves in 6-D phase space. • Consider 2-D (x and kx) and some force Fx • Electron moves a distance vx∆t in time ∆t • Electron changes momentum according to • i.e., transport between position states is taken to be classical • Transport between momentum states can be classical and quantum mechanical (scattering) • Probability of occupancy of the 2 states must be equal • Do Taylor expansion 3 Sec. 5.2 BTE: a balance equation What is being balanced here? Make the BTE into a general balance equation What does each term mean physically? 4 Sec. 5.2.1 Method of moments • The BTE solves for f, the non-equilibrium distribution function • This is very difficult to do. • We use the Method of Moments to solve for properties related to f • Specifically, charge, current density, and KE density Note the ORDER of progression of the exponent of v. 5 Sec. 5.2 Method of moments Multiply each term by Cvn and integrate over k. For example, zeroth order moment: This gives Why zero for R and G? What is this well-known equation? How do we find Je in order to evaluate n ? 6 Sec. 5.2.2 1st order moment What is the meaning of div JΦ ? What are W and <<τM>> ? We now have an expression for Je What is the difficulty in evaluating Je ? 7 Sec. 5.2.2 Continuity of current density • Write down the steady-state version of the current continuity equation • Neglect the gradient in KE/electron • This gives us the DDE • and gives definitions for mobility and diffusivity.